aa r X i v : . [ m a t h . AG ] J un ON THE DERIVED CATEGORY OF THE IWAHORI-HECKEALGEBRA
EUGEN HELLMANN
Abstract.
We state a conjecture that relates the derived category of smoothrepresentations of a p -adic split reductive group with the derived category of(quasi-)coherent sheaves on a stack of L-parameters. We investigate the con-jecture in the case of the principal block of GL n by showing that the functorshould be given by the derived tensor product with the family of representa-tions interpolating the modified Langlands correspondence over the stack ofL-parameters that is suggested by the work of Helm and Emerton-Helm. Contents
1. Introduction 22. Spaces of L-parameters 62.1. Basic properties 72.2. Derived categories of (quasi-)coherent sheaves 133. Smooth representations and modules over the Iwahori-Hecke algebra 153.1. Categories of smooth representations 163.2. The main conjecture 183.3. A generalization of the conjecture 214. The case of GL n GL Introduction
We study the smooth representation theory of a split reductive group G overa non-archimedean local field F . The classification of the irreducible smooth G -representations is one of the main objectives of the local Langlands program. Oneaims to parametrize these representations by so called L-parameters, together withsome additional datum (a representation of a finite group associated to the L-parameter). Such a parametrization has been established in the case of GL n ( F ) .For split reductive groups it has been established by Kazhdan and Lusztig for thoseirreducible smooth representations of G that have a non-trivial fixed vector underan Iwahori subgroup I ⊂ G , see [19]. In this case an L-parameter just becomes aconjugacy class of ( ϕ, N ) , where ϕ is a semi-simple element of the Langlands dualgroup ˇ G , and N ∈ Lie ˇ G , satisfying Ad( ϕ )( N ) = q − N . Here q is the number ofelements of the residue field of F . This parametrization depends on an additionalchoice, called a Whittaker datum.In this paper we formulate a conjecture that lifts the Langlands classification toa fully faithful embedding of the category Rep( G ) of smooth G -representations (onvector spaces over a field C of characteristic zero) to the category of quasi-coherentsheaves on the stack of L-parameters. It turns out that this conjecture has tobe formulated on the level of derived categories. As one of the main tools in thestudy of smooth representations is parabolic induction, we ask this fully faithfulembedding to be compatible with parabolic induction in a precise sense. Moreover,the conjectured functor should depend on the choice of a Whittaker datum.The conjecture can be made more precise in the case of the principal BernsteinBlock Rep [ T, ( G ) of Rep( G ) , i.e. the block containing the trivial representation.This block coincides with the full subcategory Rep I G of smooth G -representationsgenerated by their I -fixed vectors for a choice of an Iwahori-subgroup I ⊂ G . As Rep I G is equivalent to the category of modules over the Iwahori-Hecke algebra theconjecture comes down to a conjecture about the derived category of the Iwahori-Hecke algebra.In the main part of the paper we investigate the conjecture in the case of G = GL n ( F ) and the principal block by relating it to the construction of a familyof G -representations interpolating the (modified) local Langlands correspondence,following the work of Emerton-Helm [11].We describe the conjecture and our results in more detail. Fix a finite extension F of Q p , or of F p (( t )) . Let G be a split reductive group over F and write G = G ( F ) .We fix a field C of characteristic zero and shall always assume that contains asquare root q / of q . We denote by ˇ G the dual group of G , considered as areductive group over C . More generally, for every parabolic (or Levi) P (or M ) of G we will write P = P ( F ) (respectively M = M ( F ) ) for its group of F -valued pointsand ˇ P (respectively ˇ M ) for its dual group over C . For each parabolic subgroup P ⊂ G with Levi M (normalized) parabolic induction defines a functor ι GP from M -representations to G -representations.On the other hand we denote by X WDˇ G the space of Weil-Deligne representationswith values in ˇ G , that is, the space whose C -valued points are pairs ( ρ, N ) consistingof a smooth representation W F → ˇ G ( C ) of the Weil group W F of F and N ∈ Lie ˇ G satisfying the usual relation Ad( ρ ( σ ))( N ) = q −|| σ || N, N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 3 where || − || : W F → Z is the projection. We shall write [ X WDˇ G / ˇ G ] for the stackquotient by the obvious ˇ G -action.Let us write Z ( ˇ G ) for the global sections of the structure sheaf on [ X WDˇ G / ˇ G ] ,or equivalently the coordinate ring of the GIT quotient X WDˇ G // ˇ G . Moreover, wewrite Z ( G ) for the Bernstein center of Rep( G ) . With these notations we state thefollowing conjecture. For the sake of brevity we state the conjecture in a vague formand refer to the body of the paper for a more precise formulation of the conjecture. Conjecture 1.1.
There exists the following data: (i)
For each ( G , B , T , ψ ) consisting of a reductive group G , a Borel subgroup B ,a split maximal torus T ⊂ B , and a (conjugacy class of a) generic character ψ : N → C × there exists an exact and fully faithful functor R ψG : D + (Rep( G )) −→ D +QCoh ([ X WDˇ G / ˇ G ]) , (ii) for ( G , B , T , ψ ) as in (i) and a parabolic subgroup P ⊂ G containing B wedenote by M the Levi-quotient of P . Then the functors R ψG and R ψ M M satisfya compatibility with parabolic induction ι GP . Here ψ M is the restriction of ψ to the unipotent radical of the Borel B M of M via the splitting M → P defined by T .These data satisfy the following conditions: (a) If G = T is a split torus, then R T = R ψT is induced by the equivalence Rep( T ) ∼ = QCoh( X WDˇ T ) given by local class field theory. (b) Let ( G , B , T , ψ ) be as in (i) . The morphism Z ( ˇ G ) → Z ( G ) defined by fullyfaithfulness of R ψG is independent of the choice of ψ and induces a surjection ω G : (cid:26) Bernstein componentsof
Rep( G ) (cid:27) −→ (cid:26) connected componentsof X WDˇ G (cid:27) . (c) For ( G , B , T , ψ ) as in (i) there is an isomorphism R ψG ((c-ind GN ψ )) ∼ = O [ X WDˇ G / ˇ G ] . In this paper we mainly focus on the conjecture in the case of the principalblock of
Rep( G ) . If T ⊂ G is a split maximal torus, we write Rep [ T, ( G ) for theBernstein block of those representations π such that all irreducible subquotients of π are subquotients of a representations induced from an unramified T -representation.Then parabolic induction restricts to a functor ι GP : Rep [ T M , ( M ) → Rep [ T, ( G ) for any choice of maximal split tori T ⊂ G and T M ⊂ M (as the categories do notdepend on these choices).On the other hand we denote by X ˇ G = { ( ϕ, N ) ∈ ˇ G × Lie ˇ G | Ad( ϕ )( N ) = q − N } the space of L-parameters (corresponding to the representations in the principalblock) and write [ X ˇ G / ˇ G ] for the stack quotient by the action of ˇ G induced byconjugation. We obtain similar spaces [ X ˇ P / ˇ P ] etc. for parabolic subgroups P ⊂ G (or their Levi quotients). If T is a (maximal split) torus, then X ˇ T is just the dualtorus ˇ T .In this case the relation between the the Bernstein center Z G = Z [ T, ( G ) ofthe category Rep [ T, ( G ) and the GIT quotient X ˇ G // ˇ G can be made precise: the EUGEN HELLMANN center Z G can naturally be identified with the functions on the adjoint quotient of ˇ G and hence Z G acts on categories of modules over X ˇ G as well as on Rep [ T, ( G ) .The following conjecture is a slightly more precise version of Conjecture 1.1 in thecase of the block Rep [ T, ( G ) . Equivalently, the conjecture can be interpreted as aconjecture about the derived category of the Iwahori-Hecke algebra, and we shalltake this point of view in the last part of the paper when we discuss the case of GL n . In the case of modules over an affine Hecke algebra (where q is an invertibleindeterminate) a similar conjecture is considered in (ongoing) work of Ben-Zvi–Helm-Nadler. Conjecture 1.2.
There exists the following data: (i)
For each ( G , B , T , ψ ) consisting of a reductive group G a Borel subgroup B ,a maximal split torus T ⊂ B and a (conjugacy class of a) generic character ψ : N → C × there exists an exact and fully faithful Z G -linear functor R ψG : D + (Rep [ T, ( G )) −→ D +QCoh ([ X ˇ G / ˇ G ]) , (ii) for ( G , B , T , ψ ) as in (i) and each parabolic subgroup P ⊂ G containing B there exists a natural Z G -linear isomorphism ξ GP : R ψG ◦ ι GP −→ ( Rβ ∗ ◦ Lα ∗ ) ◦ R ψ M M of functors D + (Rep [ T M , M ) → D +QCoh ([ X ˇ G / ˇ G ]) such that the various ξ GP are compatible (in a precise sense). Here M is the Levi quotient of P , thecharacter ψ M is the restriction of ψ to the unipotent radical of B M = B ∩ M (using a splitting M ֒ → P of P → M ), and α :[ X ˇ P / ˇ P ] −→ [ X ˇ M / ˇ M ] β :[ X ˇ P / ˇ P ] −→ [ X ˇ G / ˇ G ] are the morphisms on stacks induced by the natural maps ˇ P → ˇ M and ˇ P → ˇ G .For a maximal split torus T the functor R T = R ψ T T is induced by the identification Rep [ T, ( T ) ∼ = C [ T /T ◦ ] - mod ∼ = QCoh( ˇ T ) , were T ◦ ⊂ T is the maximal compact subgroup. Moreover, for ( G , B , T , ψ ) as in (i) there is an isomorphism R ψG ((c-ind GN ψ ) [ T, ) ∼ = O [ X ˇ G / ˇ G ] . In fact it turns out that in the formulation of the conjecture the stack [ X ˇ P / ˇ P ] has to be replaced by a derived variant. Again, we refer to the body of the paperfor details and a more precise formulation of the conjecture.In the case G = GL n ( F ) we consider a candidate for the conjectured func-tor. Emerton and Helm [11] have suggested (in the context of ℓ -adic deforma-tion rings rather than the stack [ X ˇ G / ˇ G ] ) the existence of a family V G of smooth G -representations on [ X ˇ G / ˇ G ] that interpolates the modified local Langlands cor-respondence. A candidate for the family V G was constructed by Helm in [16].The modified local Langlands correspondence assigns to ( ϕ, N ) ∈ X ˇ G ( C ) a cer-tain representation LL mod ( ϕ, N ) that is indecomposable, induced from a parabolicsubgroup, has a unique irreducible subrepresentation, which is a generic representa-tion, and its unique irreducible quotient is the representation LL( ϕ, N ) associated to The author was not aware of their project when coming up with the conjecture and with theresults in this paper.
N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 5 ( ϕ, N ) by the local Langlands correspondence. In the context of modules over theIwahori-Hecke algebra H G the H G -modules corresponding to the representations LL mod ( ϕ, N ) are often referred to as the standard modules .Roughly we expect that, in the GL n -case, the functor R G = R ψG should begiven by the derived tensor product with V G (we omit the superscript ψ from thenotation as in the case of GL n there is a unique Whittaker datum). For the preciseformulation it is more convenient to pass from G -representations to modules overthe Iwahori-Hecke algebra H G . The family of H G -modules associated to V G bytaking I -invariants is in fact a H G ⊗ Z G O [ X ˇ G / ˇ G ] -module M G that is coherent over O [ X ˇ G / ˇ G ] .We consider the functor(1.1) R G : D + ( H G - mod) −→ D +QCoh ([ X ˇ G / ˇ G ]) mapping π to t π ⊗ L H G M G . Here t π is π considered as a right module over H G bymeans of the standard involution H G ∼ = H op G , and we point out that the derivedtensor product can easily be made explicit, as H G has finite global dimension.Every (standard) Levi subgroup of GL n ( F ) is a product of some GL m ( F ) , and wehence can construct similar functors(1.2) R M : D + ( H M - mod) −→ D +QCoh ([ X ˇ M / ˇ M ]) for every Levi M . Over a certain (open and dense) regular locus X regˇ G of X ˇ G , seesection 2.1 for the definition, we can relate the functor R G to Conjecture 1.2 asfollows. Theorem 1.3.
Let G = GL n . For each parabolic P ⊂ G with Levi M the restrictionof (1 . to the regular locus is a Z M -linear functor R reg M : D + ( H M - mod) −→ D +QCoh ([ X regˇ M / ˇ M ]) . satisfying compatibility with parabolic induction as in Conjecture . . Moreover, R G ((c-ind GN ψ ) I [ T, ) ∼ = O [ X ˇ G / ˇ G ] for any choice of a generic character ψ : N → C × of the unipotent radical N of aBorel subgroup B ⊂ GL n ( F ) . In the case GL ( F ) we can also control the situation for non-regular ( ϕ, N ) andprove fully faithfulness: Theorem 1.4.
Let G = GL ( F ) and T ⊂ B ⊂ G denote the standard maximaltorus respectively the standard Borel. The functors R G and R T defined by (1 . arefully faithful and there is a natural Z G -linear isomorphism ξ GB : R G ◦ ι GB −→ ( Rβ ∗ ◦ Lα ∗ ) ◦ R T , where α and β are defined as in Conjecture 1.2 (ii) . We finally return to GL n for arbitrary n , but restrict to the case of ( ϕ, N ) with ϕ regular semi-simple. Over the regular semi-simple locus the situation in factcan be controlled very explicitly and we are able to compute examples. Given ( ϕ, N ) ∈ X ˇ G ( C ) with regular semi-simple ϕ we write X ˇ G, [ ϕ,N ] for the Zariski-closure of its ˇ G -orbit. Theorem 1.5.
Let ( ϕ, N ) ∈ X ˇ G ( C ) and assume that ϕ is regular semi-simple.Then R G (LL mod ( ϕ, N )) = O [ X ˇ G, [ ϕ,N ] / ˇ G ] . EUGEN HELLMANN
Moreover, in the GL n case, at least after restricting to the regular semi-simplelocus, the conjectured functor R G should be uniquely determined by the conditionsin Conjecture 1.2. After formal completion we can prove a result in that direction.For a character χ : Z G → C we write ˆ H G,χ for the completion of the Iwahori-Heckealgebra H G with respect to the kernel of χ . Similarly we can consider the formalcompletion ˆ X ˇ G,χ of X ˇ G with respect to the pre-image of (the closed point of theadjoint quotient defined by) χ in X ˇ G . Then (1 . extends to a functor ˆ R G,χ : D + ( ˆ H G,χ -mod ) −→ D +QCoh ([ ˆ X ˇ G,χ / ˇ G ]) , and similarly for (standard) Levi subgroups M ⊂ G . Theorem 1.6.
Let ϕ ∈ ˇ G ( C ) be regular semi-simple and let χ : Z G → C denotethe character defined by the image of ϕ in the adjoint quotient. The set of functors ˆ R M,χ : D + ( ˆ H M,χ - mod) −→ D +QCoh ([ ˆ X ˇ M,χ / ˇ M ]) for standard Levi subgroups M ⊂ G , is uniquely determined (up to isomorphism)by requiring that they are ˆ Z M,χ -linear, compatible with parabolic induction, and that ˆ R T,χ is induced by the identification ˆ H T,χ - mod = QCoh( ˆ X ˇ T ,χ ) . Finally, I would like to mention that I was led to Conjecture 1.2 by considerationsabout p -adic automorphic forms and moduli spaces of p -adic Galois representations.In fact we hope that the conjecture extends (in a yet rather vague sense) to a p -adic picture, which should have implications on the computation of locally algebraicvectors in the p -adic Langlands program, as in work of Pyvovarov [25], which infact inspired the computation in section 4.6. We do not pursue this direction here,but will come back to this in the future. Acknowledgments:
I very much like to thank Christophe Breuil, MichaelRapoport, Timo Richarz, Peter Schneider, Jakob Scholbach, Benjamin Schraen andMatthias Strauch for many helpful discussions and for their interest. Moreover, Iwould like to thank Johannes Anschütz and Arthur-César Le-Bras for asking manyhelpful questions and pointing out some mistakes in an earlier version of the paper.Special thanks go to Peter Scholze for his constant interest and encouragement afterI explained him my first computations.2.
Spaces of L-parameters
We fix a field C of characteristic and a prime p with power q = p r . Let G bea linear algebraic group over C and let g denote its Lie-Algebra, considered as a C -scheme. We define the C -scheme X G as the scheme representing the functor R ( ϕ, N ) ∈ ( G × g )( R ) | Ad( ϕ )( N ) = q − N } on the category of C -algebras.The scheme X G comes with a canonical G -action, by conjugation on G and bythe adjoint action on g . We write [ X G /G ] for the stack quotient of X G by thisaction. For obvious reasons this is an algebraic stack (or Artin stack). Given a ho-momorphism α : G → H of linear algebraic groups, we obtain canonical morphisms X G → X H of schemes and [ X G /G ] → [ X H /H ] of stacks. N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 7
Basic properties.
We study the basic properties of the spaces X G and [ X G /G ] . Proposition 2.1. (i) Assume that G is reductive. Then X G is a complete inter-section inside G × g and has dimension dim G .(ii) If G = GL n , then X G is reduced and the irreducible components are in bi-jection with the set of G -orbits in the nilpotent cone N G ⊂ g of G . Moreover let η = ( ϕ η , N η ) ∈ X G be a generic point of an irreducible component. Then ϕ η isregular semi-simple.Proof. (ii) This is [15, Theorem 3.2.] resp. [17, Proposition 4.2].(i) Helm’s argument from [17] directly generalizes to the case of a reductive group:as X G ⊂ G × g is cut out by dim G equations, it is enough to show that X G isequi-dimensional of dimension dim G .Let us write f : X G → g for the projection to the Lie algebra. We first claimthat f set-theoretically factors over the nilpotent cone N G ⊂ Lie G . In order to doso we choose an embedding G ֒ → GL m for some m . Then X G embeds into X GL m and given ( ϕ, N ) ∈ X G [15, Lemma 2.3] implies that N is mapped to a nilpotentelement of gl m . This implies N ∈ N G .The scheme N G is irreducible and a finite union of (locally closed) G -orbits forthe adjoint action, as G is reductive. Let Z ⊂ N G be such a G -orbit and let G Z ⊂ G × Z be the Z -group scheme of centralizers of the points in Z , i.e. thefiber G z of G Z over z ∈ Z is the centralizer of z in G . Then the translation actionmakes f − ( Z ) (if non-empty) into a right G Z -torsor. In particular in this case wehave dim f − ( Z ) = dim Z + dim Z G Z = dim Z + dim G z = dim G , where z ∈ Z isany (closed) point. The scheme X G now is the union of the locally closed subsets f − ( Z ) , where Z runs over all the G -orbits in N G . As all these locally closedsubsets (if = ∅ ) have dimension dim G , their closures are precisely the irreduciblecomponents of X G . It follows that X G is equi-dimensional of dimension dim G asclaimed. (cid:3) Remark . (a) The proof implies that the irreducible components of X G areindexed by a subset of the G -orbits in N G . We expect that the conclusion of (ii)holds true for a general reductive group, i.e. the scheme X G should be reduced, itsirreducible components should be in bijection with the G -orbits in the nilpotentcone and at the generic points of the irreducible components the element ϕ shouldbe regular semi-simple.(b) The only ingredient in the proof of (i) that uses the assumption that G isreductive is the fact that G acts with only finitely many orbits on its nilpotent cone.More precisely, let G be an arbitrary linear algebraic group and let G ֒ → GL m bea faithful representation. Then the proof of (i) works if G acts with finitely manyorbits on Lie G ∩N GL m . This is not true in general, even if G is a parabolic subgroupin GL m , see [4]: it follows from loc. cit. that this fails in the case of a Borel subgroupin GL m for m ≥ . The following example shows that also the statement of theProposition fails for Borel subgroups of GL n for n ≥ . We did not check that thisis the optimal bound. It is very likely possible that X B is not equi-dimensional if B is a Borel subgroup in GL . Example . Let r, d > and n = rd . Let B ⊂ GL n be the Borel subgroup ofupper triangular matrices and let ϕ = diag(1 , . . . , , q, . . . , q, . . . , q d − , . . . , q d − ) ∈ B ( C ) , EUGEN HELLMANN where each entry q i appears r times. Then a given element N = ( n ij ) ij ∈ Lie B satisfies N ϕ = qϕ N if and only if n ij = 0 for j / ∈ { ir + 1 , . . . , ( i + 1) r } . Scaling ϕ by multiplication with elements of the center Z ∼ = G m , we obtain aclosed embedding G m × Q d − i =1 A r ֒ → X B . The B -orbit of this closed subscheme isirreducible and of dimension dim (cid:0) G m × d − Y i =1 A r (cid:1) + dim B − Stab B ( ϕ ) = 1 + dim B + (cid:0) ( d − r − d r ( r +1)2 (cid:1) = 1 + dim B + r (cid:0) dr − (2 r + d ) (cid:1) . In particular we find that X B has an irreducible component of dimension strictlylarger than dim B if dr ≥ r + d . On the other hand X B has always an irreduciblecomponent of dimension dim B , namely B × { } ⊂ B × Lie B .Let G be a reductive group and let P be a parabolic subgroup. We will write ˜ X P for the scheme representing the sheafification of the functor:(2.1) R ( ϕ, N, g ) ∈ ( G × g )( R ) × G ( R ) /P ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ϕ, N ) ∈ X G ( R ) and ϕ ∈ g − P g,N ∈ Ad( g − )(Lie P ) This is a closed G -invariant subscheme of X G × G/P (where G acts on G/P byleft translation). Then ( ϕ, N ) ( ϕ, N, induces a closed embedding X P ֒ → ˜ X P which descends to an isomorphism(2.2) [ X P /P ] ∼ = −→ [ ˜ X P /G ] . Moreover, the canonical projection ˜ X P → X G is G -equivariant and the inducedmorphism [ ˜ X P /G ] → [ X G /G ] agrees under the isomorphism (2 . with the mor-phism [ X P /P ] → [ X G /G ] induced by P ֒ → G . The following lemma is a directconsequence of this discussion. Lemma 2.4.
Let G be a reductive group and P ⊂ G be a parabolic subgroup. Thenthe canonical map [ X P /P ] → [ X G /G ] induced by the inclusion P ֒ → G is proper. We continue to assume that G is reductive. We say that a point ( ϕ, N ) ∈ G × g is regular , if there are only finitely many Borel subgroups B ′ ⊂ G such that ϕ ∈ B ′ and N ∈ Lie B ′ , i.e. if (for one fixed choice of a Borel B ) the point ( ϕ, N ) has onlyfinitely many pre-images under (cid:26) ( ϕ, N, gB ) ∈ G × g × G/B (cid:12)(cid:12)(cid:12)(cid:12) ϕ ∈ g − Bg,N ∈ Ad( g − )(Lie B ) (cid:27) −→ G × g . As this morphism is proper and the fiber dimension is upper semi-continuous onthe source, the regular elements form a Zariski-open subset ( G × g ) reg ⊂ G × g .Similarly we can define a Zariski-open subset X reg G = X G ∩ ( G × g ) reg ⊂ X G . If P ⊂ G is a parabolic subgroup, we write ( P × Lie P ) reg = ( G × Lie G ) reg ∩ ( P × Lie P ) and X reg P = X P ∩ X reg G . Moreover, we write ˜ X reg P for the pre-image of X reg G under ˜ X P → X G . Then [ X reg P /P ] = [ ˜ X reg P /G ] as stacks and the morphism ˜ X reg P → X reg G is by construction a finite morphism. Moreover, if we write M for the Levi quotientof P , it is a direct consequence of the definition that the canonical projection [ X P /P ] → [ X M /M ] restricts to [ X reg P /P ] → [ X reg M /M ] . N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 9
Lemma 2.5.
The scheme X reg P is equi-dimensional of dimension dim P and a com-plete intersection inside ( P × Lie P ) reg . Moreover, the map ˜ X reg P → X reg G is surjec-tive and each irreducible component of ˜ X reg P dominates an irreducible component of X reg G .Proof. Following the first lines of the proof of Proposition 2.1, the first claim followsif we show that every irreducible component of X reg P has dimension at most dim P .Equivalently, we can show that every irreducible component of ˜ X reg P has dimensionat most dim G . This is a direct consequence of the fact that ˜ X reg P → X reg G is finite.It follows that every irreducible component of ˜ X reg P has dimension equal to dim G .As ˜ X reg P → X reg G is finite and X G is equi-dimensional of dimension dim G it followsthat every irreducible component of ˜ X reg P dominantes an irreducible component of X reg G .It remains to show that ˜ X reg P → X reg G is surjective. In fact we even show that ˜ X P → X G is surjective: we easily reduce to the case P = B a Borel subgroup, and,choosing an embedding G ֒ → GL n , we can reduce to the case of GL n . There we cancheck the claim on k -valued points for algebraically closed fields k , where it easilyfollows by looking at the Jordan canonical forms of ϕ and N . (cid:3) Remark . We remark that X reg P ⊂ X P is open, but not dense in general, as canbe deduced from Lemma 2.5 and Example 2.3. If G is reductive then we expectthat X reg G is dense in X G . In the case of GL n this is a consequence of Proposition2.1.If G = GL n and P ⊂ G is a parabolic subgroup, then G/P can be identified withthe variety of flags of type P . In particular we can identify ˜ X P with the varietyof triples ( ϕ, N, F ) consisting of ( ϕ, N ) ∈ X G and a ( ϕ, N ) -stable flag of type P .From now on we will often use this identification. Lemma 2.7.
Let G = GL n and let P ⊂ G be a parabolic. Then ˜ X reg P is reduced.Proof. To prove that ˜ X reg P is reduced, it remains to show that is generically reduced.Let ξ = ( ϕ ξ , N ξ , F ξ ) ∈ ˜ X reg P be a generic point. Under ˜ β P : ˜ X reg P → X G the point ξ maps to a generic point η = ( ϕ ξ , N ξ ) of X G and hence ϕ ξ is regular semi-simple.It is enough to show that ˜ β − P ( η ) is reduced. But as ϕ ξ is regular semi-simple thespace of ϕ ξ -stable flags is a finite disjoint union of reduced points. Hence its closedsubspace of flags that are in addition stable under N ξ has to be reduced as well. (cid:3) Remark . Let n ≤ and P ⊂ GL n a parabolic subgroup. We point out that theargument in Remark 2.2 (b) implies that X P is a complete intersection in P × Lie P .But if n ≥ , it is not true that every irreducible component of ˜ X P dominates anirreducible component of X GL n . Indeed, one can compute that if n = 4 and P = B is a Borel, then there is an irreducible component of ˜ X B on which the Frobenius ϕ is semi-simple with eigenvalues λ, qλ, qλ, q λ for some indeterminate λ . Thiscomponent clearly can not dominate an irreducible component of X GL . However,for n ≤ one can compute that every irreducible component of X P is the closureof an irreducible component of X reg P . In particular we deduce that X P is reducedif n ≤ . In the general case ( P ⊂ G a parabolic subgroup of a reductive group)we do not know whether X P is reduced. Remark . Let P ⊂ G be a parabolic subgroup. The morphism β P : ˜ X P → X G clearly is not flat, as its fiber dimension can jump. But even the finite morphism ˜ X reg P → X reg G is not flat: at the intersection points of two irreducible components of X reg G the number of points in the fiber (counted with multiplicity) can jump.However, if G = GL n and x = ( ϕ, N ) ∈ X reg G is a point such that ( ϕ ss , N ) is theL-parameter of a generic representation, then the argument of [1, Lemma 1.3.2.(1)]implies that X G is smooth at x . In this case miracle flatness [22, Theorem 23.1]implies that there is a neighborhood U of x in X reg G such that β − P ( U ) → U is finiteflat. Lemma 2.10.
Let G be reductive and P ⊂ G be a parabolic with Levi quotient M .(i) The morphism X reg P → X reg M has finite Tor-dimension.(ii) Let P ′ ⊂ P be a second parabolic subgroup. Let M ′ denote the Levi quotient of P ′ and P ′ M ⊂ M denote the image of P ′ in M . Then the diagrams (2.3) X P ′ / / (cid:15) (cid:15) X P (cid:15) (cid:15) and X reg P ′ / / (cid:15) (cid:15) X reg P (cid:15) (cid:15) X P ′ M / / X M X reg P ′ M / / X reg M are cartesian and the fiber product on the right hand side is Tor-independent.Proof. (i) Let U ⊂ P denote the unipotent radical of P and fix a section M ֒ → P of the canonical projection. We write u ⊂ p for the Lie algebras of U resp. P and m for the Lie algebra of M . Then we obtain a commutative diagram ( M × U ) × ( m × u ) ∼ = ψ / / * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ P × p π (cid:15) (cid:15) M × m , where the horizontal arrow ψ is induced by multiplication and the other twomorphisms are the canonical projections. Let r = dim M and s = dim U . Let I ⊂ Γ( P × Lie P, O P × Lie P ) be the ideal defining X P ֒ → P × Lie P , i.e. the idealgenerated by the entires of Ad( ϕ )( N ) − q − N , where ϕ and N are the universalelements over P resp. Lie P . Then we deduce from the diagram that I can begenerated by elements f , . . . , f r , g . . . , g s such that f , . . . , f r ∈ Γ( M × m , O M × m ) ⊂ Γ( P × Lie P, O P × Lie P ) , where ( f , . . . , f r ) is the ideal defining X M ⊂ M × m . It follows that the ideal ( g , . . . , g s ) is the ideal defining X P as a closed subscheme of π − ( X M ) ∼ = A sX M .Let us now write K ( g , . . . , g s ) for the Koszul complex defined by g , . . . , g s onthe open subscheme π − ( X M ) reg = π − ( X M ) ∩ ( P × Lie P ) reg of π − ( X M ) . This isa finite complex of flat O X M -modules and we claim that it is a resolution of O X reg P .Indeed, g , . . . , g s cut out the closed subscheme X reg P ⊂ π − ( X M ) reg which is ofcodimension s by Lemma 2.5. As π − ( X M ) reg is Cohen-Macaulay (it is an opensubscheme of an affine space over X M and X M is Cohen-Macaulay as a consequenceof Proposition 2.1) it follows from [10, Corollaire 16.5.6] that g . . . , g s is a regularsequence and hence the Koszul complex is a resolution of its -th cohomology whichis O X reg P .(ii) The fact that the squares are fiber products follows from the fact that P ′ isthe pre-image of P ′ M under P → M . We show that the square on the right isTor-independent. As in (i) we have a Koszul complex K ( g , . . . , g s ) on π − ( X M ) reg which is a O X reg M -flat resolution of O X reg P . Consider the closed embedding(2.4) X reg P ′ ֒ → π − ( X P ′ M ) ∩ ( P × p ) reg . N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 11
As the (2 . is cartesian, the restrictions of g , . . . , g s to π − ( X P ′ M ) ∩ ( P × Lie P ) reg generate the ideal defining the closed embedding (2 . and it remains to show thatthe pullback of the Koszul complex K ( g , . . . , g s ) along (2 . is a resolution of its -th cohomology group; that is we need to show that g , . . . , g s is a regular sequencein O π − ( X reg P ′ M ) ∩ ( P × Lie P ) reg ,x for all x ∈ X reg P ′ ⊂ π − ( X reg P ′ M ) ∩ ( P × Lie P ) reg . Now π − ( X reg P ′ M ) ∩ ( P × Lie P ) reg ⊂ π − ( X reg P ′ M ) ∼ = A sX reg P ′ M is an open subscheme and hence it is Cohen-Macaulay as X reg P ′ M is Cohen-Macaulayby Lemma 2.5. The claim now follows again from [10, Corollaire 16.5.6] and thefact that X reg P ′ is equi-dimensional of dimension dim π − ( X reg P ′ M ) − s . (cid:3) Example . Let us point out that the left cartesian diagram of (2 . is notnecessarily Tor-independent without restricting to the regular locus. Let us consider r = d = 3 (so that dr = 2 r + d ) in Example 2.3. Let B ⊂ GL n be the Borel subgroupof upper triangular matrices, where n = rd = 9 . Then the above example showsthat X B is not equi-dimensional, and hence the defining ideal is not generatedby a regular sequence. Let P ⊂ GL be a standard parabolic containing B withLevi M = GL × GL . Then the classification of [4] shows that P as well as theBorel B M of M have the property that they act only via finitely many orbitson Lie P ∩ N GL resp. Lie B M ∩ N M . In particular X P and X B M are completeintersections in P × Lie P resp. B M × Lie B M by Remark 2.2. As in the proof abovewe can construct generators f , . . . , f dim B of the ideal defining X B ⊂ B × Lie B such that Tor-independence of (2 . is equivalent to exactness of the Koszul complex K ( f , . . . , f dim B ) in negative degrees. Let x ∈ X B ⊂ B × Lie B be a point that lieson an irreducible component of X B of dimension strictly larger than dim B . Then(the germs of) f , . . . , f dim B lie in the maximal ideal m B × Lie
B,x ⊂ O B × Lie
B,x andthe Koszul complex defined by these elements is not exact, as they do not form aregular sequence (because O X B ,x is not equi-dimensional of dimension dim B ).We reformulate the first claim of the Lemma in terms of stacks. Corollary 2.12.
Let P ′ ⊂ P ⊂ G be parabolic subgroups of G and let M be theLevi quotient of P and P ′ M the image of P ′ in M. Then the diagram [ X P ′ /P ′ ] / / (cid:15) (cid:15) [ X P /P ] (cid:15) (cid:15) [ X P ′ M /P ′ M ] / / [ X M /M ] of stacks is cartesian.Proof. Note that P → M induces an isomorphism P/P ′ ∼ = M/P ′ M . As in (2 . we define a closed M -invariant subscheme Y P ′ M ⊂ X M × M/P ′ M as the schemerepresenting the sheafification of the functor R (cid:26) ( ϕ, N, g ) ∈ X M ( R ) × M ( R ) /P ′ M ( R ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ∈ g − P ′ M g and N ∈ Ad( g − )(Lie P ′ M ) (cid:27) . Then we have a canonical isomorphism [ X P ′ M /P ′ M ] ∼ = [ Y P ′ M /M ] . Similarly we definea P -invariant closed subscheme Y P ′ ⊂ X P ′ × P/P ′ such that [ X P ′ /P ′ ] ∼ = [ Y P ′ /P ] . Then the diagram Y P ′ / / (cid:15) (cid:15) X P (cid:15) (cid:15) Y P ′ M / / X M is cartesian by the above lemma. Let U ⊂ P denote the unipotent radical of P ,then it follows that [ Y P ′ /U ] / / (cid:15) (cid:15) [ X P /U ] (cid:15) (cid:15) Y P ′ M / / X M is cartesian diagram of stacks with M -action and with M -equivariant morphisms.The claim follows by taking the quotient by the M -action everywhere. (cid:3) The objects introduced above have variants in the world of derived (or dg-)schemes (see e.g. [9, 0.6.8] and the references cited there). The category of dg-schemes over C canonically contains the category of C -schemes as a subcategory.For any linear algebraic group G we write γ G : G × Lie G → Lie G for the morphism ( ϕ, N ) Ad( ϕ )( N ) − q − N . We denote by X G the fiber product X G / / (cid:15) (cid:15) G × Lie G γ G (cid:15) (cid:15) { } / / Lie G in the category of dg-schemes. This yields a dg-scheme X G with underlying classicalscheme cl X G = X G . If G is reductive then Proposition 2.1 implies that X G = X G .Similarly, if P ⊂ G is a parabolic subgroup of a reductive group, we denote by X reg P ⊂ X P the open sub-dg-scheme with underlying topological space X reg P . ThenLemma 2.5 implies that X reg P = X reg P is a classical scheme.For any linear algebraic group G we write [ X G /G ] for the stack quotient of X G by the canonical action of G . This is an algebraic dg-stack in the sense of [9, 1.1].Similarly to the case of schemes we can view any algebraic stack as an algebraicdg-stack. Moreover, recall that every dg-stack S has an underlying classical stack cl S .If G is reductive and P ⊂ G is a parabolic we also consider the stacks [ X reg G /G ] and [ X reg P /P ] . Then [ X G /G ] = [ X G /G ] , [ X reg G /G ] = [ X reg G /G ] and [ X reg P /P ] = [ X reg P /P ] . We recall that a morphism Y → Y of dg-stacks is schematic if for all affine dg-schemes Z and all morphisms Z → Y the fiber product Z × Y Y is a dg-scheme,see [9, 1.1.2]. A morphism of dg-schemes is called proper is the induced morphismof the underlying classical schemes is proper, and a morphism of algebraic dg-stacksis proper if the morphism of underlying classical stacks is proper in the sense of [21, In [9] dg-stacks are simply called stacks. In order to distinguish between derived and non-derived variants we will always write dg-stack.
N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 13
Definition 7.11]. Similarly to the non-derived results above we obtain the followinglemma.
Lemma 2.13.
Let G be a reductive group and let P ⊂ G be a parabolic subgroupwith Levi quotient M .(i) The morphism [ X P /P ] → [ X G /G ] is schematic and proper.(ii) Let P ′ ⊂ P be a second parabolic subgroup and write P ′ M ⊂ M for the image of P ′ in M . Then X P ′ ∼ = X P ′ M × X M X P and [ X P ′ /P ′ ] ∼ = [ X P ′ M /P ′ M ] × [ X M /M ] [ X P /P ] . Proof. (i) As in the non derived case we can rewrite [ X P /P ] as [ ˜ X P /G ] where ˜ X P ⊂ X G × G/P is the closed G -invariant sub-dg-scheme obtained by making X P ⊂ X G × { P } ⊂ X G × G/P invariant under the G -action. More precisely ˜ X P ⊂ G × Lie G × G/P can bedescribed as follows: Let us for the moment write Y ⊂ G × Lie G × G for the closedsubscheme Y = (cid:26) ( ϕ, N, g ) ∈ G × Lie G × G (cid:12)(cid:12)(cid:12)(cid:12) g − ϕg ∈ P Ad( g − ) N ∈ Lie P (cid:27) Note that there is an action of G × P on Y by ( h, p ) . ( ϕ, N, g ) = ( hϕh − , Ad( h ) N, hgp − ) . Then the canonical projection Y → G × Lie G × G/P is a G -equivariant P -torsor,while the obvious map Y → P × Lie P is a P -equivariant G -torsor. We let ˜ X ′ P ⊂ Y denote the pullback of P -equivariant closed dg-subscheme X P ⊂ P × Lie P . Byconstruction this is a G × P -equivariant closed dg-subscheme of Y which descentsto a G -equivariant closed dg-subscheme ˜ X P ⊂ X G × G/P ⊂ G × Lie G × G/P.
Now [ X P /P ] = [ ˜ X P /G ] → [ X G /G ] is obtained taking the quotient by G from thecanonical G -equivariant morphism ˜ X P → X G . The claim follows from this togetherwith the observation that cl ˜ X P = ˜ X P and cl [ X P /P ] = [ X P /P ] .(ii) This is a direct consequence of the definition of the fiber product in the categoryof dg-schemes and the fact that P ′ is the pre-image of P ′ M under the (flat) morphism P → M . (cid:3) Derived categories of (quasi-)coherent sheaves.
Given a scheme or astack X (or a derived scheme or a derived stack) we write D QCoh ( X ) for the derivedcategory of quasi coherent sheaves on X , see [9, 1.2] and denoted by QCoh( X ) inloc. cit. We write D b Coh ( X ) for the full subcategory of objects that only havecohomology in finitely many degrees which moreover is coherent.If X is a noetherian scheme then D b Coh ( X ) coincides with the full subcategoryof the derived category D ( O X -mod ) of O X -modules, consisting of those complexesthat have coherent cohomology and whose cohomology is concentrated in finitedegrees. Similarly, if X is a (classical) algebraic stack, then D b Coh ( X ) , and moregenerally D +QCoh ( X ) , agrees with the definition of the bounded derived categoryof coherent sheaves, respectively with the bounded below derived category of quasi-coherent sheaves as defined in [21]. This is true after passing to the underlying homotopy category. The derived categories in[9] are by definition ∞ -categories while the derived categories in [21] are classical triangulatedcategories. Lemma 2.14.
Let G be a reductive group and let P be a parabolic subgroup withLevi quotient M , and let α : [ X P /P ] → [ X M /M ] and β : [ X P /P ] → [ X G /G ] denote the canonical morphisms. Then the maps Lα ∗ : D QCoh ([ X M /M ]) −→ D QCoh ([ X P /P ]) Rβ ∗ : D QCoh ([ X P /P ]) −→ D QCoh ([ X G /G ]) preserve the subcategories D +QCoh ( − ) and D b Coh ( − ) .Proof. In the case of Rβ ∗ the claim directly follows from the fact that β is properand schematic. We prove the claim for Lα ∗ . As the properties of belonging to D +QCoh ( − ) or D b Coh ( − ) can be checked over the smooth cover X P of [ X P /P ] it isenough to show that pullback along the morphism α ′ : X P −→ X M preserves D +QCoh ( − ) and D b Coh ( − ) . Moreover, both properties may be checked afterforgetting the O X P -module structure, and only remembering the O P × Lie P -modulesstructure. As in the proof of Lemma 2.10 we find that O X P can be represented bya complex F • P of flat O X M -modules that are coherent as O P × Lie P -modules, and Lα ′∗ is identified with the functor − ⊗ L O X M F • P . The claim follows from this. (cid:3) Remark . (a) We point out that using the definition of the derived categoriesas in [9] has the advantage that there is a canonical pullback functor between thederived categories of quasi-coherent sheaves on stacks. At least as long as we onlyconsider non-derived stacks (as e.g. [ X G /G ] or [ X reg G /G ] ) there is a definition of thederived category of quasi-coherent sheaves in [21]. However, the definition of thepullback functor in loc. cit. meets some problems. Lemma 2.14 essentially tells usthat we could as well use the definition of [21] and only consider complexes thatare bounded below.(b) The explicit description of the pullback in the proof of Lemma 2.14 could alsobe used to completely bypass the use of derived schemes, or derived stacks. In theend we will be interested in the composition Rβ ∗ ◦ Lα ∗ rather than in the individualfunctors. Hence instead of Lα ∗ we might use the construction − ⊗ L O XM F • P andcarefully define the O [ X G /G ] -action after push-forward (and after descent to thestack quotient). However, it seems to be more natural to use derived stacks thansuch an explicit workaround.Let P ⊂ P be parabolic subgroups of a reductive group G with Levi-quotients M i , i = 1 , . We write P ⊂ M for the image of P in M . Then P ⊂ M is aparabolic subgroup with Levi quotient M . We obtain a diagram(2.5) [ X P /P ] β / / α (cid:15) (cid:15) α ! ! β * * [ X P /P ] β / / α (cid:15) (cid:15) [ X G /G ][ X P /P ] β / / α (cid:15) (cid:15) [ X M /M ][ X M /M ] where the upper left diagram is cartesian (in the category of dg-stacks). N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 15
Lemma 2.16.
In the above situation we have a natural isomorphism (2.6) ( Rβ ∗ ◦ Lα ∗ ) ◦ ( Rβ ∗ ◦ Lα ∗ ) ∼ = −→ Rβ ∗ ◦ Lα ∗ of functors D QCoh ([ X M /M ]) −→ D QCoh ([ X G /G ]) . Proof.
The upper right square is (derived) cartesian and β is schematic andproper, in particular it is quasi-compact and quasi-separated. Hence [9, Propo-sition 1.3.6, 1.3.10] implies that the natural base change morphism Lα ∗ ◦ Rβ ∗ −→ Rβ ∗ ◦ Lα ∗ is an isomorphism. We obtain the natural isomorphism ( Rβ ∗ ◦ Lα ∗ ) ◦ ( Rβ ∗ ◦ Lα ∗ ) = Rβ ∗ ◦ ( Lα ∗ ◦ Rβ ∗ ) ◦ Lα ∗ ∼ = −→ Rβ ∗ ◦ ( Rβ ∗ ◦ Lα ∗ ) ◦ Lα ∗ = Rβ ∗ ◦ Lα ∗ . (cid:3) We point out that working only with classical schemes, we still obtain a naturaltransformation between the corresponding functors: if we consider the underlyingclassical stacks in the diagram (2 . and keep the same notations for the morphismsby abuse of notation, then there still is a natural base change morphism, but it isnot necessarily an isomorphism as the fiber product might not be Tor-independent.However, it becomes an isomorphism when we restrict the functors to the regularlocus, i.e. we consider them as functors D QCoh ([ X reg M /M ]) → D QCoh ([ X reg G /G ]) , as in this case the classical and the derived picture coincide. Indeed, after therestriction to the regular locus the derived fiber product equals the classical fiberproduct by the Tor-independence in Lemma 2.10.3. Smooth representations and modules over the Iwahori-Heckealgebra
Let F be a finite extension of Q p (or of F p (( t )) ) with residue field k F and let q = p r = | k F | . In the following let G be a split reductive group over F and write G = G ( F ) . From now on we will assume that C contains a square root q / of q .We fix a choice of this root.We will always fix T ⊂ B ⊂ G a split maximal torus and a Borel subgroup.By this choice we can define the dual group ˇ G of G considered as an algebraicgroup over C . Moreover, we denote by ˇ T ⊂ ˇ B ⊂ ˇ G the dual torus, resp. thedual Borel. More generally, given a parabolic subgroup P ⊂ G containing B , wedenote by ˇ P ⊂ ˇ G the corresponding parabolic subgroup of the dual group. Wewrite W = W G = W ( G , T ) for the Weyl group of ( G , T ) . If P ⊂ G is a parabolicsubgroup containing B , then the choice of T defines a lifting of the Levi quotient M of P to a subgroup of G . Similarly, we regard the dual group ˇ M of M as a subgroupof ˇ G containing the maximal torus ˇ T . We write W M ⊂ W for the Weyl group of ( M , T ) .Let ˇ G// ˇ G denote the GIT quotient of ˇ G with respect to its adjoint action onitself. The inclusion ˇ T ֒ → ˇ G induces an isomorphism ˇ T /W ∼ = ˇ G// ˇ G . The projection X ˇ G → ˇ G induces a map(3.1) χ = χ G : X ˇ G −→ ˇ G −→ ˇ G// ˇ G = ˇ T /W which is ˇ G -equivariant and hence induces a map ¯ χ = ¯ χ G : [ X ˇ G / ˇ G ] → ˇ T /W .Similarly, we obtain morphisms χ M : X ˇ M → ˇ T /W M and ¯ χ M : [ X ˇ M / ˇ M ] → ˇ T /W M . Categories of smooth representations.
Let us write
Rep( G ) for the cat-egory of smooth representations of G on C vector spaces. It is well known that Rep( G ) has a decomposition into Bernstein blocks Rep( G ) = Y [ M,σ ] ∈ Ω( G ) Rep [ M,σ ] ( G ) , where Ω( G ) is a set of equivalence classes of a Levi M of G and a cuspidal rep-resentation σ of M , see [2, III, 2.2] for example. We will restrict our attention tothe Bernstein component Rep [ T, ( G ) , where is the trivial representation of thetorus T . Given π ∈ Rep( G ) we write π [ T, for its image under the projection to Rep [ T, ( G ) . Moreover, we will write Z G for the center of the category Rep [ T, ( G ) ,then Z G ∼ = −→ Γ( ˇ
G// ˇ G, O ˇ G// ˇ G ) , see below for an explicit description. This isomorphism allows us to identify thecategory Z G -mod of Z G -modules with the category QCoh( ˇ
T /W ) of quasi-coherentsheaves on the adjoint quotient ˇ G// ˇ G = ˇ T /W of ˇ G , and the category Z G -mod fg offinitely generated Z G -modules with the category Coh( ˇ
T /W ) of coherent sheaves on ˇ T /W . We obtain an identification of derived categories(3.2) D ( Z G -mod ) ∼ = D QCoh ( ˇ
T /W ) , D b ( Z G -mod fg ) ∼ = D b Coh ( ˇ
T /W ) . We use these identifications and the morphism ¯ χ : [ X ˇ G / ˇ G ] → ˇ T /W to make D +QCoh ([ X ˇ G / ˇ G ]) and D b Coh ([ X ˇ G / ˇ G ]) into Z G -linear categories.If P ⊂ G is a parabolic subgroup with Levi quotient M , we write ι GP = Ind GP ( δ / P ⊗ − ) : Rep( M ) −→ Rep( G ) for the normalized parabolic induction, and ι GP for normalized parabolic inductionof the opposite parabolic P of P (note that the normalization uses the choice of q / ). These functors are exact and restrict to functors Rep [ T M , ( M ) −→ Rep [ T, ( G ) (for any choice of a maximal split torus T M of M ). Using a splitting M ֒ → P ⊂ G to the projection we obtain a morphism ˇ M // ˇ M −→ ˇ G// ˇ G which is obviously independent of the choice of M ֒ → G . Then the functors ι GP and ι GP are linear with respect to the morphism Z G ∼ = Γ( ˇ G// ˇ G, O ˇ G// ˇ G ) −→ Γ( ˇ
M // ˇ M , O ˇ M// ˇ M ) ∼ = Z M , see below for details.Let us write D (Rep [ T, ( G )) respectively D + (Rep [ T, ( G )) for the derived cate-gory, respectively for the bounded below derived category, of Rep [ T, ( G ) . Moreover,we write D b (Rep [ T, , fg ( G )) for the full subcategory of complexes whose cohomol-ogy is concentrated in bounded degrees and is finitely generated as a C [ G ] -module. N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 17
Then ι GP and ι GP induce functors D (Rep [ T M , ( M )) −→ D (Rep [ T, ( G )) D + (Rep [ T M , ( M )) −→ D + (Rep [ T, ( G )) D b (Rep [ T M , , fg ( M )) −→ D b (Rep [ T, , fg ( G )) which we will also denote by ι GP respectively ι GP .Given two parabolic subgroup P ⊂ P of G with Levi quotient M respectively M . We write P for the image of P in M . Then we have natural isomorphisms(3.3) ι GP ◦ ι M P −→ ι GP ,ι GP ◦ ι M P −→ ι GP of functors D (Rep [ T M , ( M )) → D (Rep [ T, ( G )) .Finally, recall that a Whittaker datum is a G -conjugacy class of tuples ( B , ψ ) ,where B ⊂ G is a Borel subgroup and ψ : N → C × is a generic character of N = N ( F ) , where N ⊂ B is the unipotent radical. As above we fix the choice ofa Borel subgroup B and a maximal split torus T ⊂ G . For a parabolic P ⊂ G containing B with Levi quotient M we write ψ M : N M → C × for the restriction of ψ to the unipotent radical N M ⊂ N of the Borel B M = B ∩ M of M . Note that the M -conjugacy class of ( B M , ψ M ) does not depend on the choice of M ֒ → G (i.e. onthe choice of T ).We can describe the above categories of representations in terms of modulesover Iwahori-Hecke algebras. In order to do so, let us fix a hyperspecial vertex inthe apartment of the Bruhat-Tits building of G defined by the maximal torus T ,i.e. we fix O F -models of ( G , T ) . The choice of a Borel B then defines an Iwahorisubgroup I ⊂ G . We write Rep I G for the category of smooth representations π of G on C -vector spaces that are generated by their Iwahori fixed vectors π I and Rep I fg G ⊂ Rep I G for the full subcategory of representations that are finitelygenerated (as C [ G ] -modules). It is well known that the category Rep I G does notdepend on the choice of I and agrees with the Bernstein block Rep [ T, ( G ) .Let H G = H ( G, I ) = End G ( c-ind GI I ) denote the Iwahori-Hecke algebra. Then π π I = Hom G ( c-ind GI I , π ) induces an equivalence of categories between Rep [ T, ( G ) = Rep I G and the cat-egory H G -mod of H G -modules. This equivalence identifies Rep I fg G and the fullsubcategory H G -mod fg ⊂ H G -mod of finitely generated H G -modules. Moreover, itidentifies the center Z G of Rep [ T, ( G ) with the center of the Iwahori-Hecke algebra H G . Then we have an isomorphism Z G ∼ = C [ X ∗ ( T )] W = C [ X ∗ ( ˇ T )] W = Γ( ˇ T /W, O ˇ T /W ) = Γ( ˇ
G// ˇ G, O ˇ G// ˇ G ) (see for example [14, Lemma 2.3.1]), which is in fact independent of the choice ofthe Iwahori I .Given a representation π ∈ Rep I G and a Z G -module ρ we will sometimes (byabuse of notation) write π ⊗ Z G ρ for the pre-image of the H G -module π I ⊗ Z G ρ under the equivalence Rep I G ∼ = H G -mod (and similarly for corresponding derivedfunctors). Remark . Note that if G = T is a split torus, then I = I T = T ◦ is the uniquemaximal compact subgroup of T and we have canonical identifications(3.4) C [ X ∗ ( T )] ∼ = C [ T /T ◦ ] = H T . where the first isomorphism is given by µ µ ( ̟ ) for the choice of a uniformizer ̟ of F (note that this isomorphism is independent of this choice). We often usethis isomorphism to identify unramified characters and H T -modules.Let P ⊂ G be a parabolic subgroup containing B with Levi quotient M and write P = P ( F ) and M = M ( F ) . Set I M = I G ∩ M , which is an Iwahori-subgroupof M , in particular Rep [ T M , ( M ) = Rep I M M . There is a canonical embedding H M ֒ → H G such that the diagrams(3.5) Rep I M M ( − ) IM / / ι GP (cid:15) (cid:15) H M -mod Hom H M ( H G , − ) (cid:15) (cid:15) and Rep I M M ( − ) IM / / ι GP (cid:15) (cid:15) H M -mod H G ⊗ H M − (cid:15) (cid:15) Rep I G G ( − ) I / / H G -mod Rep I G G ( − ) I / / H G -mod . commute. Note that this is equivalent to the commutativity of the diagram(3.6) Rep I M M ( − ) IM / / H M -mod Rep I G G ( − ) I / / r GP ( − ) O O H G -mod . forget O O Here r GP ( − ) is the normalized Jacquet-module which is the left adjoint functor to ι GP ( − ) . It is also the right adjoint functor to ι GP ( − ) by Bernstein’s second adjointnesstheorem. By abuse of notation we will often write ι GP respectively ι GP for the functors Hom H M ( H G , − ) respectively H G ⊗ H M − on Hecke modules.The embedding H M ⊂ H G induces an embedding Z G ⊂ Z M , where Z M is thecenter of Rep [ T M , ( M ) which is identified with the center of H M , such that thecanonical diagram Z G / / (cid:15) (cid:15) Γ( ˇ
T /W, O ˇ T/W ) (cid:15) (cid:15) Z M / / Γ( ˇ
T /W M , O ˇ T /W M ) commutes. We deduce that ι GP and ι GP are Z G -linear. In particular, for a Z G -module ρ we obtain natural isomorphisms(3.7) ι GP ( − ⊗ Z G ρ ) −→ ι GP ( − ) ⊗ Z G ρ,ι GP ( − ⊗ Z G ρ ) −→ ι GP ( − ) ⊗ Z G ρ, and similarly for the corresponding functors on the derived category.3.2. The main conjecture.
Using the notations introduced above we state thefollowing conjecture. Variants of the conjecture have been around in representationtheory in the past years in (ongoing) work of Ben-Zvi–Nadler–Helm, and of X. Zhu.
Conjecture 3.2.
There exists the following data: (i)
For each ( G , B , T , ψ ) consisting of a reductive group G , a Borel subgroup B ,a split maximal torus T ⊂ B , and a (conjugacy class of a) generic character ψ : N → C × there exists an exact and fully faithful Z G -linear functor R ψG : D + (Rep [ T, ( G )) −→ D +QCoh ([ X ˇ G / ˇ G ]) , N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 19 (ii) for ( G , B , T , ψ ) as in (i) and each parabolic subgroup P ⊂ G containing B there exists a natural Z G -linear isomorphism ξ GP : R ψG ◦ ι GP −→ ( Rβ ∗ ◦ Lα ∗ ) ◦ R ψ M M of functors D + (Rep [ T M , M ) → D +QCoh ([ X ˇ G / ˇ G ]) . Here M is the Levi quo-tient of P and α : [ X ˇ P / ˇ P ] −→ [ X ˇ M / ˇ M ] ,β : [ X ˇ P / ˇ P ] −→ [ X ˇ G / ˇ G ] are the morphisms on stacks induced by the natural maps ˇ P → ˇ M and ˇ P → ˇ G .These data satisfy the following conditions: (a) If G = T is a split torus, then R T = R ψT is induced by the identification (3 . and viewing a sheaf on ˇ T as an ˇ T -equivariant sheaf with the trivial ˇ T -action (note that ˇ T acts trivially on ˇ T = X ˇ T ). (b) Let ( G , B , T , ψ ) as in (i) and let P ⊂ P ⊂ G be parabolic subgroups con-taining B with Levi quotients M and M . Let P denote the image of P in M . Then, with the notations from (2 . , the diagram R ψG ◦ ι GP ξ GP u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ( . ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ Rβ , ∗ Lα ∗ ◦ R ψ M M ( . ) (cid:15) (cid:15) R ψG ◦ ι GP ◦ ι M P ξ GP (cid:15) (cid:15) ( Rβ , ∗ Lα ∗ ) ◦ ( Rβ , ∗ Lα ∗ ) ◦ R ψ M M ξ M P / / ( Rβ , ∗ Lα ∗ ) ◦ R ψ M M ◦ ι M P is a commutative diagram of functors D + (Rep [ T M , ( M )) −→ D +QCoh ([ X ˇ G / ˇ G ]) . (c) For any ( G , B , T , ψ ) as in (i) let (c-ind GN ψ ) [ T, denote the projection of thecompactly induced representation c-ind GN ψ to Rep [ T, ( G ) . Then R ψG ((c-ind GN ψ ) [ T, ) ∼ = O [ X ˇ G / ˇ G ] . Let us point out that the Z G -linearity of the conjectured functor R ψG implies thatfor each ρ ∈ D + ( Z G - mod) there is a natural isomorphism ψ G,ρ : R ψG ( − ⊗ L Z G ρ ) ∼ = −→ R ψG ( − ) ⊗ L O [ X ˇ G/ ˇ G ] L ¯ χ ∗ G ρ of functors D + (Rep [ T, ( G )) → D +QCoh ([ X ˇ G / ˇ G ]) which is functorial in ρ (in theobvious sense). Moreover, given P ⊂ G as in (ii), the Z G -linearity of the naturalisomorphism ξ GP implies that the natural transformations ψ M,ρ and ψ G,ρ are com-patible with parabolic induction. We do not spell this out explicitly in terms ofcommutative diagrams.
Remark . (a) We expect that the conjectured functor R ψG induces a functor D b (Rep [ T, , fg ( G )) −→ D b Coh ([ X ˇ G / ˇ G ]) . This would allow to extend the functor to the full derived category D (Rep [ T, ( G )) :as Rep [ T, , fg ( G ) ∼ = H G -mod fg and as H G has finite global dimension, see [2, 4. Theorem 29], the full derived category D (Rep [ T, ( G )) is the ind-completion of D b (Rep [ T, , fg ( G )) . Hence the conjectured functor would extend to a fully faithfuland exact functor D (Rep [ T, ( G )) −→ IndCoh([ X ˇ G / ˇ G ]) , where IndCoh([ X ˇ G / ˇ G ]) is the ind-completion of D b Coh ([ X ˇ G / ˇ G ]) . Note that this cat-egory differs from D QCoh ([ X ˇ G / ˇ G ]) , as X ˇ G is singular. However, there is a canonicalequivalence IndCoh + ([ X ˇ G / ˇ G ]) ∼ = −→ D +QCoh ([ X ˇ G / ˇ G ]) , see e.g. [9, 3.2.4]. In particular, restricting to bounded below objects, the conjecturethat the (yet hypothetical) functor R ψG is fully faithful does not depend on whetherwe consider it as a functor with values in IndCoh + ([ X ˇ G / ˇ G ]) or with values in D +QCoh ([ X ˇ G / ˇ G ]) . We hence arrive with a conjecture that parallels the formulationof the geometric Langlands program, see [12]. Also the conjectured compatibilitywith parabolic induction agrees with the compatibility with parabolic induction inloc. cit.(b) Recall that an L-parameter for G that is trivial on inertia is a ˇ G -conjugacy class [ ϕ, N ] of ( ϕ, N ) ∈ X ˇ G ( C ) with ϕ semi-simple. We write S [ ϕ,N ] = C [ ϕ,N ] /C ◦ [ ϕ,N ] forthe quotient of the centralizer of ( ϕ, N ) by its connected component of the identity.By the classification of Kazhdan-Lusztig [19, Theorem 7.12] the irreducible repre-sentations in Rep I G (respectively the simple objects in H G -mod) are in bijection with pairs ([ ϕ, N ] , ρ ) , where [ ϕ, N ] is an L-parameter and ρ is an irreducible rep-resentation of S [ ϕ,N ] . This parametrization depends on an additional choice thatcorresponds to the choice of a Whittaker datum ( B , ψ ) . More precisely, the classi-fication in [19] (which in the case of GL n coincides with the Bernstein-Zelevinskyclassification [3]) associates to ([ ϕ, N ] , ρ ) an indecomposable representation (respec-tively Hecke module) π ψ [ ϕ,N ] ,ρ which has a unique irreducible quotient. Conjecture3.2 should have the following relation with this classification. Given [ ϕ, N ] let uswrite X ◦ ˇ G, [ ϕ,N ] ⊂ X ˇ G for the locally closed subscheme whose C -valued points are given by those ( ϕ ′ , N ′ ) such that [ ϕ, N ] is the ˇ G -conjugacy class of ( ϕ ′ ss , N ) , where ϕ ′ ss is the semi-simplification of ϕ ′ . Moreover, we denote by X ˇ G, [ ϕ,N ] = X ◦ ˇ G, [ ϕ,N ] its Zariski closure. We assume that ϕ is regular semi-simple. Given an irreduciblerepresentation ρ of S [ ϕ,N ] on a finite dimensional C -vector space, we can use ρ todefine a ˇ G -equivariant coherent sheaf ˜ F [ ϕ,N ] ,ρ ∈ Coh( X ˇ G, [ ϕ,N ] ) which hence defines a coherent sheaf F [ ϕ,N ] ,ρ on the closed substack [ X ˇ G, [ ϕ,N ] / ˇ G ] ⊂ [ X ˇ G / ˇ G ] . We then expect that the conjectured functor R G has the property R ψG ( π ψ [ ϕ,N ] ,ρ ) = F [ ϕ,N ] ,ρ . At least for classical groups this is true. Otherwise one needs to add an assumption on therepresentation ρ . N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 21
If the L-parameter [ ϕ, N ] is generic, there is a unique ψ -generic representation π inthe L-packet defined by [ ϕ, N ] . With the above notations this representation is therepresentation π = π ψ [ ϕ,N ] , trivial . Then, the expected formula above specializes to R ψG ( π ) = O [ X ˇ G, [ ϕ,N ] / ˇ G ] (c) We point out that the conjectured functor R ψG will not be essentially surjective.In fact this is already obvious in the case G = T a split torus. Here R T = R ψT isthe derived version of the functor H T - mod ∼ = QCoh( ˇ T ) −→ QCoh([ ˇ
T / ˇ T ]) . The morphism on the right hand side is the embedding given by equipping a quasi-coherent sheaf with the trivial ˇ T -action. Obviously ˇ T -equivariant sheaves withnon-trivial ˇ T -action are not in the essential image.There is also a second obstruction for essential surjectivity. Let [ ϕ, N ] be anL-parameter such that ϕ is semi-simple but not regular semi-simple. Then (usingthe notation of (b)) the structure sheaf of the closed substack [ X ssˇ G, [ ϕ,N ] / ˇ G ] ⊂ [ X ˇ G, [ ϕ,N ] / ˇ G ] of pairs ( ϕ ′ , N ′ ) where ϕ ′ is (pointwise) semi-simple should not be in the essentialimage of the functor R G .(d) Finally we point out that in the conjecture it is necessary to pass to derivedcategories. Heuristically this can be explained by the fact that flat morphismson the representation theory side correspond to non-flat morphisms on the side ofstacks: for example H G is flat over its center, whereas the canonical morphism ¯ χ G : [ X ˇ G / ˇ G ] −→ ˇ T /W is not flat (as it maps some irreducible components to proper closed subschemes of ˇ T /W ). Moreover, we will see below that in the case of GL n ( F ) the trivial repre-sentation will be mapped to a complex concentrated in cohomological degree − n ,see Remark 4.37 below. Hence, without passing to derived categories, the functorcan not be fully faithful. The canonical t-structures on the source (respectivelytarget) should correspond to an exotic t-structure on the other side. However, wehave no idea how this t-structure could be described intrinsically. Moreover, theformulation of the conjecture needs the passage to derived schemes respectivelyderived stacks: as parabolic induction is transitive (in the sense that (3 . is anisomorphism), the base change morphism (2 . has to be an isomorphism as well.However, in the world of classical schemes the corresponding cartesian diagram isnot Tor-independent in general.3.3. A generalization of the conjecture.
Conjecture 3.2 in fact is a special caseof a more general conjecture about the category
Rep( G ) , instead of the Bernsteinblock Rep [ T, ( G ) . Let us describe this generalization.We continue to assume that G is a split reductive group with dual group ˇ G . Letus write W F for the Weil group of F and I F ⊂ W F for the inertia group. Wedefine the space of ˇ G -valued Weil-Deligne representations to be the scheme X WDˇ G representing the functor R (cid:26) ρ : W F → ˇ G ( C ) , N ∈ Lie ˇ G (cid:12)(cid:12)(cid:12)(cid:12) ρ | J is trivial for some J ⊂ I F open Ad( ρ ( σ ))( N ) = q −|| σ || N (cid:27) on C -algebras R . Here ||− || : W F → Z is the usual projection. It is easy to see that X WDˇ G is a union of affine schemes and is equipped with a ˇ G -action via conjugationon ρ and via the adjoint action on N . Spaces of Weil-Deligne representations like X WDˇ G are studied in (ongoing) work of Dat, Helm, Kurinczuk and Moss in the(more complicated) case where the coefficients are in Z [ p ] rather than the field C of characteristic zero.Similarly, for every parabolic subgroup ˇ P ⊂ ˇ G we can define the scheme X WDˇ P and the derived scheme X WDˇ P that come equipped with ˇ P -actions.The inclusion ˇ P ֒ → ˇ G and the projection ˇ P → ˇ M onto the Levi-quotient ˇ M of ˇ P induce morphisms(3.8) β WDˇ P : [ X WDˇ P / ˇ P ] −→ [ X WDˇ G / ˇ G ] ,α WDˇ P : [ X WDˇ P / ˇ P ] −→ [ X WDˇ M / ˇ M ] of the respective stack quotients. Moreover, we will write X WDˇ G // ˇ G for the GITquotient of X WDˇ G by the ˇ G -action. As in the case of the space of ( ϕ, N ) -modules X ˇ G it is easy to show that β WDˇ P is proper. The following conjecture summarizesexpected properties of the spaces just introduced. Conjecture 3.4.
Let ˇ P ⊂ ˇ G be a parabolic subgroup with Levi-quotient ˇ M .(i) The space X WDˇ G is a local complete intersection.(ii) The morphism α WDˇ P : [ X WDˇ P / ˇ P ] → [ X WDˇ M / ˇ M ] has finite Tor-dimension.(iii) There is a unique morphism X WDˇ M // ˇ M → X WDˇ G // ˇ G making the diagram [ X WDˇ P / ˇ P ] β WDˇ P y y ssssssssss α WDˇ P % % ❑❑❑❑❑❑❑❑❑ [ X WDˇ G / ˇ G ] (cid:15) (cid:15) [ X WDˇ M ˇ M ] (cid:15) (cid:15) X WDˇ G // ˇ G X
WDˇ M // ˇ M o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ commutative.Remark . In the case of the space of ( ϕ, N ) -modules X ˇ G all these propertieshave been verified in section 2. Indeed, for (iii) we are left to remark that themorphism [ X ˇ G / ˇ G ] −→ X ˇ G // ˇ G is just the morphism ¯ χ from (3 . , i.e. the GIT quotient X ˇ G // ˇ G agrees with theadjoint quotient ˇ G// ˇ G . This can be seen as follows: the morphism ϕ ( ϕ, definesa closed embedding ˇ G ֒ → X ˇ G which is the inclusion of an irreducible component.As ˇ G is reductive and C has characteristic the category of ˇ G -representations issemi-simple and we obtain a closed embedding ˇ G// ˇ G −→ X ˇ G // ˇ G. As source and target are reduced (as ˇ G and X ˇ G are) it is enough to show that themorphism is bijective. This comes down to proving that for ( ϕ, N ) ∈ X ˇ G ( k ) , foran algebraically closed field k , there exists ϕ ′ ∈ ˇ G ( k ) such that ˇ G · ( ϕ ′ , ∩ ˇ G · ( ϕ, N ) = ∅ . By (the proof of) Lemma 2.5 we may assume that ϕ ∈ ˇ B and N ∈ Lie ˇ B for someBorel ˇ B ⊂ ˇ G . Let G m act on X ˇ G by the sum of the positive roots, then the closure N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 23 of G m · ( ϕ, N ) contains in addition the point ( ϕ ′ , for some ϕ ′ ∈ ˇ G such that ϕ and ϕ ′ have the same image in the adjoint quotient ˇ G// ˇ G .Let us write Z ( G ) for the Bernstein center of the category Rep( G ) . Given aBernstein component Ω of Rep( G ) we denote its center by Z Ω ( G ) . Moreover, wedenote by Z ( ˇ G ) = Γ( X WDˇ G // ˇ G, O X WDˇ G // ˇ G ) the ring of functions on the GIT quotient X WDˇ G // ˇ G . If X ⊂ X WDˇ G is a connectedcomponent, we write Z X ( ˇ G ) for the ring of functions on the GIT quotient X// ˇ G . Remark . If G = T is a split torus, then the isomorphism F × → W ab F of localclass field theory identifies X WDˇ T with the scheme representing the functor R ρ : F × −→ ˇ T ( R ) smooth character } on the category of C -algebras. In particular, the scheme X WDˇ T decomposes into adisjoint union of copies of ˇ T indexed by the smooth characters O × F → ˇ T ( C ) . Thisdecomposition induces an equivalence of categories(3.9) Rep( T ) ∼ = QCoh( X WDˇ T ) . Assuming the geometric properties of stacks of L-parameters conjectured above,we are able to state a generalization of Conjecture 3.2.
Conjecture 3.7.
There exists the following data: (i)
For each ( G , B , T , ψ ) consisting of a reductive group G , a Borel subgroup B ,a split maximal torus T ⊂ B , and a (conjugacy class of a) generic character ψ : N → C × there exists an exact and fully faithful functor R ψG : D + (Rep( G )) −→ D +QCoh ([ X WDˇ G / ˇ G ]) , (ii) for ( G , B , T , ψ ) as in (i) and each parabolic subgroup P ⊂ G containing B there exists a natural isomorphism ξ GP : R ψG ◦ ι GP −→ ( Rβ WDˇ
P , ∗ ◦ Lα WD , ∗ ˇ P ) ◦ R ψ M M of functors D + (Rep( M )) → D +QCoh ([ X WDˇ G / ˇ G ]) . Here M is the Levi quo-tient of P and α WDˇ P and β WDˇ P are the morphisms defined in (3 . .These data satisfy the following conditions: (a) If G = T is a split torus, then R T = R ψT is induced by the equivalence (3 . given by local class field theory. (b) Let ( G , B , T , ψ ) be as in (i) . The morphism Z ( ˇ G ) → Z ( G ) defined by fullyfaithfulness of R ψG is independent of the choice of ψ and induces a surjection ω G : (cid:26) Bernstein componentsof
Rep( G ) (cid:27) −→ (cid:26) connected componentsof X WDˇ G (cid:27) . (c) Let ( G , B , T , ψ ) and P be as in (ii) . Then the natural isomorphism ξ GP is Z ( ˇ G ) -linear for the Z ( ˇ G ) -linear structure on Rep( M ) defined by themorphism Z ( ˇ G ) −→ Z ( ˇ M ) −→ Z ( M ) that is given by the composition of the morphism Z ( ˇ G ) → Z ( ˇ M ) inducedby Conjecture . (iii) with the morphism Z ( ˇ M ) → Z ( M ) of (b) . (d) Let ( G , B , T , ψ ) as in (i) and let P ⊂ P ⊂ G be parabolic subgroups con-taining B with Levi quotients M and M . Let P denote the image of P in M . Then the diagram R ψG ◦ ι GP ξGP t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ( ∗ ) ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ Rβ WDˇ P , ∗ Lα WD , ∗ ˇ P ◦ R ψM M ∗∗ ) (cid:15) (cid:15) R ψG ◦ ι GP ◦ ι M P ξGP (cid:15) (cid:15) ( Rβ WDˇ P , ∗ Lα WD , ∗ ˇ P ) ◦ ( Rβ WDˇ P , ∗ Lα WD , ∗ ˇ P ) ◦ R ψM M ξM P / / ( Rβ WDˇ P , ∗ Lα WD , ∗ ˇ P ) ◦ R ψM M ◦ ι M P is a commutative diagram of functors D + (Rep( M )) −→ D +QCoh ([ X WDˇ G / ˇ G ]) . Here ( ∗ ) is the natural isomorphism given by transitivity of parabolic in-duction and ( ∗∗ ) is a base change isomorphism defined by the analogousdiagram as in (2 . . (e) For ( G , B , T , ψ ) as in (i) there is an isomorphism R ψG (c-ind GN ψ ) ∼ = O [ X WDˇ G / ˇ G ] . Remark . (a) It should be possible to construct the expected morphism Z ( ˇ G ) −→ Z ( G ) of (b) in the conjecture, without referring to the conjectured functor R ψG . Theconstruction of morphisms like that is part of the work of Dat, Helm, Kurinczukand Moss mentioned above. In the case of GL n a result like this has been establishedby Helm and Moss [18].(b) In fact Z ( ˇ G ) should coincide with the stable Bernstein center as defined byHaines in [13, 5.3.]. This should be a rather direct consequence of the definitions,but we did not check the details. Then the morphism Z ( ˇ G ) → Z ( G ) of (b) inthe Conjecture should coincide with the morphism constructed in [13, Proposition5.5.1] assuming the local Langlands correspondence.Let us point out that the morphism ω G from (b) can not be expected to be abijection in general, as not every Bernstein component Ω is ψ -generic in the senseof [6, 4.3] (note that the notions of being ψ -generic and being simply ψ -generic of[6] agree by Example 4.5 (1) of loc. cit., as G is assumed to be (quasi-)split). Moreprecisely, Conjecture 3.7 predicts that the restriction of ω G induces a bijection (cid:26) ψ -generic Bernsteincomponents of Rep( G ) (cid:27) −→ (cid:26) connectedcomponents of X WDˇ G (cid:27) , and that for a ψ -generic Bernstein component Ω the induced morphism(3.10) Z ω G (Ω) ( ˇ G ) −→ Z Ω ( G ) is an isomorphism. Indeed, combining 4.2. Corollary and 4.3. Theorem of [6] wededuce that the ψ -generic components are precisely those components Ω such that (c-ind GN ψ ) Ω = 0 . Moreover, the morphism (3 . fits in the commutative diagram End G ((c-ind GN ψ ) Ω ) ∼ = / / End [ X WDˇ G, Ω / ˇ G ] ( O [ X WDˇ G, Ω / ˇ G ] ) Z Ω ( G ) ∼ = O O Z ω G (Ω) ( ˇ G ) , ∼ = O O o o N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 25 where X WDˇ G, Ω ⊂ X WDˇ G denotes the connected component defined by ω G (Ω) . Herethe upper horizontal arrow is an isomorphism by (e) and fully faithfulness in theconjecture, the right vertical arrow is an isomorphism by definition and the leftvertical arrow is an isomorphism by [6, 4.3. Theorem]. Remark . In the case G = GL n there is (up to conjugation) a unique choice of ( B , ψ ) and every Bernstein component of Rep(GL n ( F )) is ψ -generic, see e.g. [6, 4.5,Examples (2)]. Moreover, in this case one can show that X WDˇ G decomposes into adisjoint union X WDGL n = a n X n , where n = ( n [ τ ] ) is a tuple of non-negative integers n [ τ ] indexed by the W F -conjugacy classes [ τ ] of irreducible I F -representations τ : I F → GL d τ ( C ) suchthat n = X [ τ ] [ W F : W τ ] · n τ d τ . Here W τ ⊂ W F is the W F -stabilizer of a representative τ of [ τ ] . Moreover, each X n is connected and decomposes into a product where each factor is a space of ( ϕ, N ) -modules for a finite extension F ′ of F . On the other hand, the local Langlandscorrespondence for GL n ( F ) induces a bijection W F -conjugacy classes ofirreducible smooth representations τ : I F → GL m ( C ) , m ≥ ←→ equivalence classes ofcuspidal representations GL r ( F ) , r ≥ where two cuspidal representations are said to be equivalent if they differ by thetwist by an unramified character. Hence we obtain a bijection between the Bern-stein components of Rep(GL n ( F )) and the connected components of X WDGL n . Byresults of Bushnell-Kutzko [7] every Bernstein component of Rep(GL n ( F )) can bedescribed by a semi-simple type and the corresponding Hecke-algebra is in fact iso-morphic to a tensor product of Iwahori-Hecke algebras. This corresponds to thedecomposition of the connected components X n of X WDGL n into a product of spacesof ( ϕ, N ) -modules. In fact, in the case of GL n type theory and a closer inspec-tion of these decompositions should reduce Conjecture 3.7 to Conjecture 3.2 (inthe case of GL r for various r ). In particular it should be possible to generalize allresults proven in the following section for the block Rep [ T, (GL n ( F )) to the wholecategory Rep(GL n ( F )) . 4. The case of GL n In this section we consider the group G = GL n ( F ) and make Conjecture 3.2more explicit in this case. We will provide a candidate for the conjectured functorand prove that it satisfies compatibility with parabolic induction on the dense opensubset of regular elements. In the case of GL we give a full proof of the conjecture.We fix G = GL n and choose the canonical integral model of G over O F corre-sponding to the maximal compact subgroup K = GL n ( O F ) of G . In particular weassume that the hyperspecial vertex defined by K is contained in the apartmentdefined by the maximal split torus T ⊂ GL n , and I ⊂ K . We use this to obtaincanonical integral models for the choice of a Borel B ⊃ T and for parabolic sub-groups P ⊃ B as well as for their Levi quotients. We will use the same symbols forthese integral models. We will often simply write Z = Z G for the Bernstein centerof the category Rep [ T, ( G ) . The modified Langlands correspondence.
We recall the construction ofthe modified local Langlands correspondence defined by Breuil and Schneider in [5,4], see also [11, 4.2]. We restrict ourselves to the Bernstein block
Rep [ T, ( G ) .Let ̟ be a uniformizer of F . For any field extension L of C and λ ∈ L × we write unr λ : F × → L × for the unramified character mapping ̟ to λ . More generally,for λ = ( λ , . . . , λ n ) ∈ ( L × ) n we write unr λ = unr λ ⊗ · · · ⊗ unr λ n : T → L × for the unramified character of the torus T = ( F × ) n whose restriction to the i -thcoordinate is unr λ i .Write |−| = unr q − : F × → C × for the unramified character such that | ̟ | = q − .Let L be a field extension of C and let ( ϕ, N ) ∈ X ˇ G ( L ) be a ( ϕ, N ) -module suchthat ϕ is semi-simple. Then Breuil and Schneider associate to ( ϕ, N ) a smooth,absolutely indecomposable representation LL mod ( ϕ, N ) of GL n ( F ) with coefficientsin L as follows:Fix an algebraic closure ¯ L of L . Given a scalar λ ∈ ¯ L × and r ≥ let Sp( λ, r ) denote as usual the ( ϕ, N ) -module structure on ¯ L r = ¯ Le ⊕ . . . ¯ Le r − defined by(4.1) ϕ ( e i ) = q − i λN ( e i ) = ( e i +1 , i < r − , i = r − . Let
St( λ, r ) denote the generalized Steinberg representation of GL r ( F ) with coeffi-cients in ¯ L , i.e. the unique simple quotient of ι GB (unr λ ⊗ unr λ |−|⊗· · ·⊗ unr λ |−| n − ) .Given some ( ϕ, N ) ∈ X ˇ G ( L ) with ϕ semi-simple we decompose (after enlarging L if necessary) ( L n , ϕ, N ) ∼ = s M i =1 Sp( λ i , r i ) and define LL mod ( ϕ, N ) as the unique L -model of the ¯ L representation(4.2) ι GP (cid:0) St( λ , r ) ⊗ · · · ⊗ St( λ s , r s ) (cid:1) . Here P is the block upper triangular parabolic whose Levi is the block diagonalsubgroup GL r ×· · ·× GL r s and the λ i are ordered so that they satisfy the conditionof [20, Definition 1.2.4]. Remark . Note that the normalization we use differs from the one in [5] and [11].In loc. cit. the representation LL mod ( ϕ, N ) is modified by the twist by | det | − ( n − / .This has the advantage that the resulting GL n ( F ) representation has a uniquemodel over L , without assuming the existence (or fixing a choice) of q / , asproven in [5, Lemma 4.2]. As we have fixed a choice q / in the base field C ,and hence a choice of | det | − ( n − / , their argument also implies that our represen-tation LL mod ( ϕ, N ) has a (unique) model over L . The reason for these two differentnormalizations is the following: In [5] the representation should be canonically de-fined over L , without choosing q / , and moreover, in [11] the representations should(conjecturally) satisfy some local-global compatibility. In our case we work purelylocally and we are aiming for a compatibility with normalized parabolic induction.More precisely, we need Lemma 4.3 below to be true as stated (i.e. not a twistedversion of it). Anyway, the definition of normalized parabolic induction forces usto choose a square root q / .If ( ϕ, N ) ∈ X ˇ G with non semi-simple ϕ , we write LL mod ( ϕ, N ) = LL mod ( ϕ ss , N ) .Moreover, if ( ϕ, N ) is such that LL mod ( ϕ, N ) is absolutely irreducible (that is if N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 27 the ˇ G -conjugacy class [ ϕ ss , N ] is a generic L-parameter), we usually just write LL( ϕ, N ) instead of LL mod ( ϕ, N ) . Note that in this case LL( ϕ, N ) ∨ ∼ = LL(( ϕ, N ) ∨ ) ,as normalized parabolic induction commutes with contragredients and as in thiscase the parabolic induction of the contragredient representation still satisfies thecondition of [20, Definition 1.2.4]. Lemma 4.2.
Let x = ( ϕ x , N x ) ∈ X ˇ G . Then, using the notation of (3 . the center Z acts on the representation LL mod (( ϕ, N ) ∨ ) ∨ via the character χ x : Z → k ( x ) defined by χ ( x ) ∈ ˇ T /W = Spec Z Proof.
By definition of LL mod we may assume that ϕ is semi-simple. The represen-tation LL mod (( ϕ, N ) ∨ ) ∨ embeds into ι GB (unr λ ⊗ · · · ⊗ unr λ n ) for some ordering λ = ( λ , . . . , λ n ) of the eigenvalues of ϕ . Hence it follows that (cid:0) LL mod (( ϕ, N ) ∨ ) ∨ (cid:1) I embeds into H G ⊗ H T unr λ and it is enough to prove that Z ⊂ H G acts on H G ⊗ H T unr λ as asserted. But as Z ⊂ H T is the center of H G , itacts on H G ⊗ H T unr λ via the same character as on unr λ . The claim follows fromthis. (cid:3) Recall that for a regular semi-simple endomorphism ϕ of an L -vector space L n with eigenvalues in L there is a canonical bijection(4.3) { ϕ -stable complete flags F of L n } ←→ { orderings of the eigenvalues of ϕ } . If F is a flag corresponding to an ordering λ = ( λ , . . . , λ n ) of the eigenvalues of ϕ , we denote by unr F = unr λ the L -valued unramified character defined by thisordering. Lemma 4.3.
Let x = ( ϕ x , N x ) ∈ X ˇ G with ϕ x regular semi-simple and let L be an(algebraic) extension of k ( x ) containing the eigenvalues of ϕ x . Then r GB (LL mod (( ϕ x , N x ) ∨ ) ∨ ⊗ k ( x ) L ) = M F unr F , where the direct sum runs over all flags of L n stable under ϕ x and N x .Proof. The lemma is an application of the geometrical Lemma [3, 2.11, p. 448]describing the composition of parabolic induction with the Jacquet functor.Assume first that ( ϕ, N ) ⊗ k ( x ) L = Sp( λ, r ) , see (4 . . Then we need to computethe Jacquet-module of the generalized Steinberg representation St( λ, r ) = ι GB (cid:0) δ − / B ⊗ unr λ | − | ( n − / (cid:1)(cid:14) X B ( P ⊆ G ι GP (cid:0) δ − / P ⊗ unr λ | − | ( n − / (cid:1) . Here we view unr λ | − | ( n − / as a character of M for any (standard) Levi M .Computing r GB ( ι GP ( δ − / ⊗ unr λ | − | ( n − / )) using the geometrical lemma of [3] itfollows that r GB (St( λ, r )) = unr q − ( r − λ ⊗ · · · ⊗ unr q − λ ⊗ unr λ . This is the character corresponding to the ordering λq − ( r − , . . . , λq − λ, λ , i.e. tothe ordering defined by the unique ( ϕ, N ) -stable flag of Sp( λ, r ) .In the general case we decompose ( ϕ, N ) ⊗ k ( x ) L = L si =1 St( λ i , r i ) and write LL mod ( ϕ x , N x ) ⊗ k ( x ) L = ι GP (cid:0) St( λ , r ) ⊗ · · · ⊗ St( λ s , r s ) (cid:1) as in (4 . . Then again the geometrical lemma of [3] computes that its Jacquet-module is the desired one, and the claim follows from compatibility with contra-gredients. (cid:3) Remark . For C = C the lemma can be interpreted as a consequence of theclassification of Kazhdan-Lusztig [19] using equivariant K -theory, or its formulationusing Borel-Moore homology in [8, 8.1]. For ( ϕ, N ) ∈ X ˇ G ( C ) as in the lemma thefiber ˜ β − B ( ϕ, N ) of ˜ β ˇ B : ˜ X ˇ B → X ˇ G is identified with the ϕ -fixed points B ϕn ofthe variety B N of N -stable complete flags, compare [8, 8.1]. Using the inductiontheorem [19, 6] one can deduce that the H G -module LL mod ( ϕ, N ) is precisely thestandard module constructed in [8, Definition 8.1.9](note that the group C ( ϕ, N ) ofloc. cit. is trivial in the GL n -case). However, as ϕ is regular semi-simple the variety B ϕn is a finite union of points, namely the complete ( ϕ, N ) -stable flags. Hence itsBorel-Moore homology is the direct sum of copies of C indexed by these points.By construction the Hecke algebra H T acts on this direct sum as asserted in thelemma.4.2. The work of Helm and Emerton-Helm.
Emerton and Helm [11] proposedthe existence of a family of G -representations over a deformation space of ℓ -adic Ga-lois (or Weil-Deligne) representations that interpolates the modified local Langlandscorrespondence in a certain sense. A candidate for such a family was constructedin subsequent work of Helm [16]. Rather than working over ℓ -adic deformationrings we want to work with the stacks of L-parameters defined above. We reviewthe work of Emerton-Helm and Helm in this set up in order to construct a familyof G -representation on the stack [ X ˇ G / ˇ G ] .In this section we need to work with families of admissible smooth representations,compare [11, 2.1.]. We make precise what we mean by this. Let A be a noetherian C -algebra and let V be a finitely generated A [ G ] -module. We say that V is anadmissible smooth family of G representations over A , if the G -representation on V is smooth and if V K ′ is a finitely generated A -module for every compact opensubgroup K ′ ⊂ G .Denote by N ⊂ B the unipotent radical and let ψ : N → C × be a genericcharacter. Recall that an irreducible G -representation π is called generic if thereexists an embedding π ֒ → Ind GN ψ . Equivalently, π is generic if there is a surjection c-ind GN ψ ։ π .We write (c-ind GN ψ ) [ T, for the image of the compactly induced representation c-ind GN ψ in the Bernstein component Rep [ T, ( G ) = Rep I G . As in the case of GL n a Whittaker datum is unique up to isomorphism, this representation (up toisomorphism) does not depend on the Whittaker datum ( B, ψ ) .Recall that we have fixed K = GL n ( O F ) ⊃ I and consider the induced repre-sentation Ind KI I . By [26] this induced representation decomposes into a directsum(4.4) Ind KI I = M P σ ⊕ m P P in Helm’s integral ℓ -adic set up, the construction of the candidate in [16] is not complete, butdepends on a conjecture about the action of the Bernstein center (Conjecture 7.5. of [16]). Thisconjecture was proven by Helm and Moss [18]. In out set up of representations on characteristic , and only considering the Bernstein block defined by [ T, this conjecture becomes much easierand boils down to Lemma 4.2 above. N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 29 indexed by the set of partitions P of the positive integer n which is partially ordered,see [26, p. 169], and has a unique minimal element P min and a unique maximalelement P max . Let st = st G = σ P min denote the finite dimensional Steinberg rep-resentation. This representation occurs with multiplicity m P min = 1 . As c-ind GK st lies in the Bernstein component [ T, it carries a natural action of Z .We further recall from [3, 3.2, 3.5] the definition of the r -th derivative V ( r ) of a GL n ( F ) -representation V which is a smooth representation of GL n − r ( F ) . In par-ticular V ( n ) is just a C -vector space. By [16, p.5, (2)] there is natural isomorphism(4.5) Hom G (c-ind GN ψ, V ) ∼ = V ( n ) . If = v ∈ V ( n ) and V lies in the Bernstein component [ T, , then the morphismdefined by v obviously factors through (c-ind GI ψ ) [ T, .The following theorem is a summary of the results in [16, §§3,4] (translated tothe easier situation considered here). Theorem 4.5.
Let π be one of the representations (c-ind GN ψ ) [ T, and c-ind GK st .Then π is a smooth Z -representation and the n -th derivative π ( n ) is a free Z -modulesof rank . Moreover, let p ∈ Spec Z then (a) the representation π ⊗ k ( p ) is a direct sum of finite length representations. (b) the cosocle cosoc( p ) of π ⊗ k ( p ) is absolutely irreducible and generic. (c) the representation π ⊗ k ( p ) / cosoc( p ) does not contain any generic subquo-tient.Finally, the representation c-ind GK st is admissible as a Z -representation.Proof. We cite the proof from [16]. All references in this proof refer to loc. cit.In Helm’s situation the coefficients are W ( k ) for a finite field k , instead of thecharacteristic zero field C in our case. The arguments literally do not change inour set-up; except for one argument, where the classification of irreducible, smoothmod ℓ representations in terms of parabolic induction has to be replaced by thecorresponding classification of irreducible, smooth representations in characteristiczero.The case of (c-ind GI ψ ) [ T, follows from Lemma 3.2 and Lemma 3.4. In the case π = c-ind GK st admissibility follows from Theorem 4.1, and part (a) is Lemma 4.2.Properties (b) and (c) are proven in Proposition 4.9. Finally the claim on π ( n ) isCorollary 4.10.The proof of Helm’s Proposition 4.9 uses the classification of irreducible smoothmod ℓ representations of GL n ( F ) in terms of parabolic induction, and has to bereplaced by the usual Bernstein-Zelevinsky classification of irreducible, smooth rep-resentations in characteristic zero [28]. With this change of reference the proof in[16] literally does not change. (cid:3) Corollary 4.6.
There is an isomorphism of Z [ G ] -modules. (c-ind GN ψ ) [ T, ∼ = c-ind GK st , unique up to a scalar in Z × . Note that the partial ordering used here is the opposite to the standard ordering of partitions,compare [26, 3.]. Here the maximal element is given by · · · + 1 and the minimal elementis n . Proof.
By Theorem 4.5 the n -th derivative (c-ind GK st) ( n ) is locally free of rank over Z . As Z ∼ = C [ X , . . . , X n − , X ± n ] every line bundle on Spec Z is trivial andhence (c-ind GK st) ( n ) ∼ = Z . By the discussion preceding Theorem 4.5, a choice of abasis vector (which is unique up to a scalar in Z × ) gives rise to a morphism α : (c-ind GN ψ ) [ T, −→ c-ind GK st . We claim that α is an isomorphism.First we show that α is surjective: let W denote the cokernel of α . Then W isgenerated by its Iwahori fixed vectors W I and, by admissibility of c-ind GK st , the Z -module W I is finitely generated.As ( − ) I is an exact functor W I ⊗ k ( p ) = ( W ⊗ k ( p )) I and hence W = 0 if and onlyif W ⊗ k ( p ) = 0 for all p ∈ Spec Z .As α by definition induces an isomorphism α ( n ) : (c-ind GN ψ ) ( n )[ T, −→ (c-ind GK st) ( n ) and as the functor ( − ) ( n ) is exact (see e.g. [3, 3.2, Proposition]) it follows that W ( n ) = 0 and ( W ⊗ k ( p )) ( n ) = 0 for all p ∈ Spec Z . Assume that W ⊗ k ( p ) = 0 . As W ⊗ k ( p ) is a quotient of c-ind GK st ⊗ k ( p ) , Theorem 4.5 (b),(c) implies that thereexists a non-zero morphism c-ind GN ψ −→ W ⊗ k ( p ) , contradicting ( W ⊗ k ( p )) ( n ) = 0 Now c-ind GK st is projective as a G -representation and hence the surjection α hasa splitting (c-ind GN ψ ) [ T, ∼ = c-ind GK st ⊕ W ′ . As α induces an isomorphism after applying the n -th derivative ( − ) ( n ) it followsthat ( W ′ ) ( n ) = 0 . By the adjointness property (4 . is follows that the canonicalprojection β : (c-ind GN ψ ) [ T, −→ W ′ is zero and hence W ′ = 0 , as β is surjective. (cid:3) Following [16] we construct a family V G of G -representations on [ X ˇ G / ˇ G ] thatconjecturally interpolates the modified local Langlands correspondence (see Con-jecture 4.8 below for the precise meaning). Rather than constructing V G directlyon [ X ˇ G / ˇ G ] we construct a family ˜ V G on X ˇ G =: Spec A ˇ G that is ˇ G -equivariant andhence descents to [ X ˇ G / ˇ G ] . Lemma 4.7.
Let x = ( ϕ x , N x ) ∈ X ˇ G . There exists a canonical surjection (c-ind GN ψ ) [ T, ⊗ Z k ( x ) −→ LL mod (( ϕ ss x , N x ) ∨ ) ∨ that is unique up to scalar.Proof. This follows from the argument in the proof of [16, Theorem 7.9], usingLemma 4.2 instead of Conjecture 7.5. of loc. cit.. (cid:3)
Let η = ( ϕ η , N η ) ∈ X ˇ G be a generic point. Then LL( ϕ η , N η ) = LL mod ( ϕ η , N η ) = LL mod (( ϕ η , N η ) ∨ ) ∨ is an irreducible generic representation. We obtain a morphism (c-ind GN ψ ) [ T, ⊗ Z A ˇ G −→ (c-ind GN ψ ) [ T, ⊗ Z k ( η ) −→ LL( ϕ η , N η ) , where the second morphism is the choice of a surjection as in Lemma 4.7. N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 31
Let η i , i ∈ I denote the generic points of X ˇ G = Spec A ˇ G . We define ˜ V G to be the(admissible smooth) family of G -representations over A ˇ G that is the image of themorphism(4.6) (c-ind GN ψ ) [ T, ⊗ Z A ˇ G −→ Y i ∈ I LL( ϕ η i , N η i ) . Up to isomorphism, this image does not depend on the choice of the surjection (c-ind GN ψ ) [ T, ⊗ Z k ( η i ) → LL( ϕ η i , N η i ) . By abuse of notation we also write ˜ V G forthe corresponding sheaf on X ˇ G .It can easily be seen that ˜ V G is a ˇ G -equivariant quotient of (c-ind GN ψ ) [ T, ⊗ Z A ˇ G (equipped with the obvious ˇ G -equivariant structure). Hence ˜ V G descents to a quasi-coherent sheaf V G on [ X ˇ G / ˇ G ] that carries an action of G . Conjecturally this familyinterpolates the modified local Langlands correspondence: Conjecture 4.8 (compare [11]) . Let x = ( ϕ, N ) ∈ X ˇ G be any point, then ( ˜ V G ⊗ k ( x )) ∨ ∼ = LL mod (( ϕ, N ) ∨ ) . Idempotents in the Iwahori-Hecke algebra.
We will describe the familyof Hecke modules associated to the Emerton-Helm family V G in the next subsection,and relate this construction to Conjecture 3.2. Before we do so, we need somepreparation about idempotent elements in the Iwahori-Hecke algebra.Let J ⊂ G be a compact open subgroup and ( λ, W ) be a smooth representa-tion of J on a finite dimensional C -vector space with contragredient representation ( λ ∨ , W ∨ ) . Then we have a natural identification of C -algebras(4.7) End G (c-ind GJ λ ) ∼ = compactly supported f : G → End C ( W ∨ ) such that f ( j gj ) = λ ∨ ( j ) ◦ f ( g ) ◦ λ ∨ ( j ) for all g ∈ G, j , j ∈ J , where, as usual, the algebra structure on the right hand side is given by convolution.Given f ∈ H ( G, λ ) one defines ˇ f : g f ( g − ) ∨ ∈ End C ( W ) . Then f ˇ f inducesan isomorphism of C -algebras H ( G, λ ) ∼ = H ( G, λ ∨ ) op . Recall that H G = End G (c-ind GI I ) = End G (c-ind GK V ) , where V = Ind KI I .From now on we write λ for the K -representation on V . As in (4 . the representa-tion V decomposes as a direct sum of the representations isomorphic to σ P . Notethat V = Ind GL n (k) B ( k ) B ( k ) , where B ( k ) ⊂ GL n ( k ) is the special fiber of the Borel subgroup and K acts viathe quotient map K → GL n ( k ) . For a partition P we write Σ P ⊂ V for the σ P -isotypical component of V . In particular we have Σ P ∼ = σ m P P . The direct summand c-ind GK Σ P of c-ind GK V = c-ind GI I defines an idempotent element e P ∈ H G . If P = P min we will usually write e st (or e G, st if we need to refer to the group G )instead of e P min . Further we usually write e K = e P max , which is identified with thecharacteristic function of K .Using the description of the Hecke algebra (4 . the idempotent elements e P canbe described as follows. Let f P : V ∨ → V ∨ denote the endomorphism that is theidentity on Σ ∨P and zero on Σ ∨P ′ for P ′ = P . Then the idempotent element e P isdefined by e P : g ( , g / ∈ Kλ ∨ ( g ) ◦ f P = f P ◦ λ ∨ ( g ) , g ∈ K .
Note that the representation V = Ind GK I and the irreducible representations σ P are self-dual. In the case of V this follows from the computation of the smooth dualof an induced representation. In particular, the canonical identification I = ( I ) ∨ gives a canonical isomorphism α : V → V ∨ . In the case of σ P we proceed bydescending induction: the claim is obviously true for K = σ P max and for each P we can find some (integral model of a) parabolic subgroup P ⊂ GL n such that Ind GL n ( k ) P ( k ) ∼ = σ P ⊕ M P(cid:22)P ′ = P σ ⊕ a P′ P ′ for some integers a P ′ . As the induced representation on the left hand is self-dualso must be σ P .It follows that we can identify H G = H G ( V, λ ) with H G ( V, λ ∨ ) . In particular weobtain a canonical isomorphism H G ∼ = H op G . Lemma 4.9.
Let P be a partition. Then ˇ e P = e P .Proof. The canonical isomorphism α allows us to identify End C ( V ∨ , V ∨ ) with End C ( V, V ) and H G = H ( G, λ ) with H ( G, λ ∨ ) . By definition ˇ e P is the element ( g e P ( g − ) ∨ ) ∈ H ( G, λ ∨ ) = H ( G, λ ) under this identification. We calculate that ˇ e P ( g ) = ( , g / ∈ Kf ∨P ◦ ( λ ∨ ( k − )) ∨ = f ∨P ◦ λ ( k ) , g ∈ K. Here f ∨P is the idempotent endomorphism of V defined by the direct summand Σ P .As the σ P are self-dual the isomorphism α maps Σ P to Σ ∨P . Hence we concludethat (under the identification End C ( V ∨ , V ∨ ) = End C ( V, V ) using α ) the element ˇ e P equals e P . (cid:3) Recall that H G contains the finite Hecke algebra H G, = C ∞ c ( I \ K/I ) = { f : B ( k ) \ GL n ( k ) / B ( k ) → C } = (cid:26) f : K → End C V (cid:12)(cid:12)(cid:12)(cid:12) f ( k kk ) = λ ( k ) ◦ f ( k ) ◦ λ ( k ) for all k, k , k ∈ K (cid:27) = End K ( V ) as a subalgebra. This algebra contains the idempotent elements e P . Further recallthat for a parabolic subgroup P ⊂ G containing B we have an embedding H M ֒ → H G of Hecke algebras, where M = M ( F ) is the Levi of P . If P = B this gives anembedding C [ X ∗ ( T )] = H T ֒ → H G . By [14, Lemma 1.7.1] the morphism(4.8) H T ⊗ C H G, −→ H G induced by multiplication is an isomorphism of C -vector spaces. Lemma 4.10. (i) The canonical inclusion H T e G, st ⊂ H G e G, st is an equality.Moreover this module is free of rank with basis e G, st as an H T -module.(ii) Let P ⊂ G be a parabolic as above and let M = M ( F ) ⊂ G be the correspondingLevi subgroup. The isomorphism H M e M, st = H T e M, st −→ H G e G, st = H T e G, st of free H T -modules of rank defined by e M, st e G, st is an H M -module homomor-phism. N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 33
Proof. (i) It directly follows from (4 . that H T e G, st is free of rank as an H T -module. Moreover, note that H G, e G, st = (st G ) B ( k ) is a -dimensional C -vector space. This implies that f ∈ H G can be written as f = f e st + f (1 − e st ) with f ∈ H T and f ∈ H G . It follows that f e G, st = f e G, st + f (1 − e G, st ) e G, st = f e G, st ∈ H T e G, st . (ii) As the inclusion H T e G, st ⊂ H G e G, st is an equality, we also have an equality H M e G, st = H G e G, st . Therefore it is enough to show that the H M -module homo-morphism H M −→ H M e G, st mapping to e G, st factors through H M → H M e M, st . That is, we need to show (1 − e M, st ) e G, st = 0 in H G . We can check this equality in the subalgebra H G, .Translating the claim back to representation theory it comes down to the claimthat Ind GL n ( k ) P ( k ) st M ⊂ Ind GL n ( k ) P ( k ) (cid:0) Ind M ( k ) B M ( k ) (cid:1) = Ind GL n ( k ) B ( k ) contains the direct summand st G , where B M = B ∩ M is a Borel in M . This istrue, as st G is the only constituent of the right hand side that does not occur inany parabolically induced representation for a parabolic strictly larger than B . (cid:3) Corollary 4.11.
Let x = ( ϕ x , N x ) ∈ X ˇ G with ϕ x regular semi-simple and let L bean extension of k ( x ) containing all the eigenvalues of ϕ x . Then (cid:0) (c-ind GN ψ ) [ T, ⊗ Z k ( x ) (cid:1) I ∼ = ( H G e G, st ) ⊗ Z k ( x ) and after extending scalars to L its Jacquet-module is given by r GB ((c-ind GN ψ ) [ T, ⊗ Z L ) = M F unr F , where the sum is indexed by the ϕ x -stable flags F of L n . Moreover, the kernel ofthe quotient map of H T -modules (4.9) ( H G e G, st ) ⊗ Z L = M ϕ x - stable F unr F −→ M ( ϕ x ,N x ) - stable F unr F is H G -stable and the induced H G -module structure on the quotient identifies theright hand side with the I -invariants of (the scalar extension to L of ) the quotient LL mod (( ϕ x , N x ) ∨ ) ∨ in Lemma 4.7.Proof. The first claim is a direct consequence of (c-ind GN ψ ) [ T, ∼ = c-ind GK st G andthe identification (c-ind GK st G ) I = H G e G, st . The claim on the Jacquet-module follows from H G e G, st = H T e G, st and (3 . .For the second part, note that the right hand side in (4 . is uniquely determinedas an H T -module, as the characters unr F are pairwise distinct. Hence it is enoughto prove that the quotient (cid:0) (c-ind GN ψ ) [ T, ⊗ Z k ( x ) (cid:1) I −→ (cid:0) LL mod (( ϕ x , N x ) ∨ ) ∨ (cid:1) I given by Lemma 4.7 induces this quotient map on the underlying H T -modules(and after extending scalars to L ). This is a consequence of the computation of r GB (LL mod (( ϕ x , N x ) ∨ ) ∨ ⊗ k ( x ) L ) , see Lemma 4.3. (cid:3) We finish this subsection by recalling some easy facts about the passage from leftto right modules over H G . Given a left H G -module π one can view π as a right H G -module via the isomorphism H G ∼ = H op G . We write t π for this right modulestructure on π . Lemma 4.12.
Let M ⊂ G be a Levi and let π be a left H M -module. Then there isa canonical and functorial isomorphism of right H G -modules t ( H G ⊗ H M σ ) ∼ = t σ ⊗ H M H G , where the H G -module structure on the right hand side is given by right multiplica-tion.Proof. It is easily checked that ϕ ⊗ v v ⊗ ˇ ϕ defines the desired isomorphism. (cid:3) Lemma 4.13.
Let π be an H G -module and let e ∈ H G be an idempotent element.(i) There is a canonical equality of Z -modules Hom H G ( H G e, π ) = eπ = e H G ⊗ H G π. (ii) There is a canonical identification t ( H G e ) = ˇ e H as H G right modules.(iii) Let P be a partition. Then Hom H G ( H G e P , π ) = t ( H G e P ) ⊗ H G π. (iv) For two partitions P , P ′ we have e P H G e P ′ ∼ = Z m P m P′ . Proof.
Part (i) and (ii) are obvious, and (iii) is a direct consequence of (i), (ii) and ˇ e P = e P . Finally we find e P H G e P ′ = Hom H G ( H G e P , H G e P ′ ) = Hom G (c-ind GK Σ P , c-ind GK Σ P ′ )= Hom G (c-ind GK σ P , c-ind GK σ P ′ ) m P m P′ . Now (iv) follows from [23, Theorem 1.4]. (cid:3)
The Hecke-module of the interpolating family.
In subsection 4.2 weconstructed a family of G -representations V G on the stack [ X ˇ G / ˇ G ] . Let M G = ( V G ) I denote the corresponding module over the Iwahori-Hecke algebra. We write ˜ M G for the corresponding ˇ G -equivariant sheaf of O X ˇ G ⊗ Z H G -modules on X ˇ G .Given a Levi-subgroup M ⊂ G we continue to write A ˇ M for the coordinate ringof X ˇ M . Recall that we have embeddings H M ֒ → H G and a canonical isomorphism H T = Z T ∼ = A ˇ T . We consider the following commutative diagram. ˜ X regˇ Bγ y y ssssssssss (cid:2) (cid:2) ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ ˜ β B t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ X regˇ G (cid:15) (cid:15) X regˇ G × ˇ T /W ˇ T (cid:15) (cid:15) β ′ o o ˇ T /W ˇ T . o o N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 35
Lemma 4.14.
The morphism γ : ˜ X regˇ B −→ X regˇ G × ˇ T /W ˇ T is a closed immersion.Proof. Clearly γ is a finite morphism. Hence it is enough to show that γ inducesan injection on k -valued points, for algebraically closed fields k , and a surjectionon complete local rings.Let k be an algebraically closed extension of C and let ( A, m ) be a local Artinian C -algebra with residue field k . Let ( ϕ, N ) ∈ X regˇ G ( A ) and let λ , . . . , λ n ∈ A . Thenwe have to show that there is at most one complete flag F • of A n stable under ϕ and N such that ϕ acts on F i / F i − by multiplication with λ i .Assume first A = k . We prove the claim by induction on n . The case n = 1 istrivial. Assume the claim is true for n − . Then it is enough to show that thereis a unique ( ϕ, N ) -stable line F in k n on which ϕ acts by multiplication with λ .Obviously this forces F ⊂ ker N and we need to show that the ϕ -eigenspace in ker N of eigenvalue λ is one dimensional. However, if this is not the case thenthere are infinitely many pairwise distinct ( ϕ, N ) -stable lines in k n , and each canbe completed to a complete ( ϕ, N ) -stable flag. This contradicts the regularity of ( ϕ, N ) .Now assume that ( A, m ) is a general Artinian C -algebra with residue field k .Again it suffices to show that there is a unique ( ϕ, N ) -stable A -line in A n , suchthat the quotient of A n by this line is free, on which ϕ acts as multiplication by λ . By induction on the length of A we can reduce to the following situation: thereexists f ∈ A such that m f = 0 , and if A ′ = A/ ( f ) and ( ϕ ′ , N ′ ) is the image of ( ϕ, N ) in X regˇ G ( A ′ ) , then there is a unique ( ϕ ′ , N ′ ) stable A ′ -line in A ′ n on which ϕ ′ actsby multiplication with λ mod ( f ) . Let ( ¯ ϕ, ¯ N ) ∈ X regˇ G ( k ) denote the reduction of ( ϕ, N ) modulo m and let ¯ λ ∈ k denote the reduction of λ . Then the multiplicationwith f induces an embedding of k n ֒ → A n of ( ϕ, N ) -modules with cokernel A ′ n .Assume that F = Ae and F ′ = Ae ′ are two ( ϕ, N ) -stable A -lines on which ϕ actsby multiplication with λ . Then the assumption implies e ′ = αe + f v for some α ∈ A × and v ∈ k n . Let ¯ e ∈ k n denote the reduction of e modulo m , then itremains to show v ∈ k ¯ e . As ϕ ( e ) = λ e and ϕ ( e ′ ) = λ e ′ we deduce ¯ ϕ ( v ) = ¯ λ v .The discussion of the case of an algebraically closed field k above implies that it isenough to prove that v ∈ ker ¯ N . However, we assume that F and F ′ are definedby points ( ϕ, N, F • ) , ( ϕ, N, F ′• ) ∈ X regˇ G ( A ) . As X regˇ G is reduced by Lemma 2.7 andas N is nilpotent we deduce that N ( F ) = N ( F ′ ) = 0 and hence N ( f v ) = 0 whichimplies ¯ N ( v ) = 0 . (cid:3) We use the lemma to identify ˜ X regˇ G with a closed subscheme Y regˇ G of X regˇ G × ˇ T /W ˇ T .We denote by Y ˇ G ⊂ X ˇ G × ˇ T /W ˇ T = Spec( A ˇ G ⊗ Z A ˇ T ) the closure of Y regˇ G equipped with its canonical scheme structure (which is thereduced structure, as ˜ X regˇ B is reduced). Let us write ˜ A ˇ G for the correspondingquotient of A ˇ G ⊗ Z A ˇ T and β : Y ˇ G → X ˇ G for the canonical projection.We can use Lemma 4.10 to equip A ˇ G ⊗ Z A ˇ T = A ˇ G ⊗ Z H T ∼ = A ˇ G ⊗ Z H T e G, st = A ˇ G ⊗ Z H G e G, st with an H G -module structure. Proposition 4.15. (i) The kernel of the canonical morphism A ˇ G ⊗ Z A ˇ T → ˜ A ˇ G isstable under the action of H G .(ii) There is a canonical isomorphism ˜ M G ∼ = β ∗ O Y ˇ G of ˇ G -equivariant O X ˇ G ⊗ Z H G -modules.Proof. (i) By Lemma 2.5 and Lemma 2.7 the scheme Y ˇ G is reduced and every irre-ducible component of Y ˇ G dominates an irreducible component of X ˇ G . In particularthe canonical morphism ˜ A ˇ G −→ Y η ˜ A ˇ G ⊗ k ( η ) = Y η Γ( ˜ β − B ( η ) , O ˜ β − B ( η ) ) is an injection. Here the product runs over all generic points η of X ˇ G . It is thereforeenough to prove that for all generic points η of X ˇ G the kernel of the canonical map k ( η ) ⊗ Z H G e G, st = k ( η ) ⊗ Z H T e G, st = k ( η ) ⊗ Z A ˇ T −→ Γ(( ˜ β − B ( η ) , O ˜ β − B ( η ) ) is stable under the H G -action. This follows from Corollary 4.11 applied to thegeneric point η = ( ϕ η , N η ) .(ii) Consider the diagram (c-ind GN ψ ) I [ T, ⊗ Z A ˇ G / / / / ∼ = (cid:15) (cid:15) Γ( X ˇ G , ˜ M G ) (cid:31) (cid:127) / / Q η LL( ϕ η , N η ) I ∼ = (cid:15) (cid:15) A ˇ T ⊗ Z A ˇ G / / / / ˜ A ˇ G (cid:31) (cid:127) / / Q η Γ( ˜ β − B ( η ) , O ˜ β − B ( η ) ) , where the left vertical arrow comes from the identification of (c-ind GN ψ ) I [ T, = H G e G, st = H T e G, st ∼ = A ˇ T and the right vertical arrow comes from the identification of the Jacquet-moduleof LL( ϕ η , N η ) in Corollary 4.11. By construction the diagram is a commutativediagram of A ˇ G ⊗ Z H ˇ G -modules and moreover all morphisms are compatible withthe ˇ G -action. Hence these morphisms induce a canonical isomorphism Γ( X ˇ G , ˜ M G ) ∼ = ˜ A ˇ G as claimed. (cid:3) As a consequence we can easily deduce Conjecture 4.8 for regular semi-simplepoints.
Corollary 4.16.
Let x = ( ϕ, N ) ∈ X ˇ G with ϕ regular semi-simple. Then ( ˜ V G ⊗ k ( x )) ∨ ∼ = LL mod (( ϕ, N ) ∨ ) . Proof.
It follows from the proof of Proposition 4.15 that ((c-ind GN ψ ) [ T, ⊗ Z k ( x )) I −→ ˜ M G ⊗ k ( x ) ∼ = Γ( ˜ β − ( x ) , O ˜ β − ( x ) ) is a surjection of H T ⊗ k ( x ) -modules. The claim now follows from Corollary 4.11. (cid:3) Remark . Proposition 4.15 gives a canonical isomorphism ˜ M G ∼ = β ∗ O Y ˇ G of O X ˇ G ⊗ Z H G -modules which restricts to an isomorphism ˜ M G | X regˇ G ∼ = R ˜ β reg B, ∗ O ˜ X regˇ B = R ˜ β reg B, ∗ O ˜ X regˇ B N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 37 on the regular locus. Here we write ˜ β reg for the restriction of the canonical mor-phism ˜ β : ˜ X ˇ B → X ˇ G to the regular locus, and use that ˜ β reg is affine. In fact wewould like to have an identification(4.10) ˜ M G = R ˜ β B, ∗ O ˜ X ˇ B of O X ˇ G ⊗ Z H G -modules. In particular we would like to lift the H T = A ˇ T -actionon O ˜ X ˇ B to an action of H G . In the regular case this is precisely the content of theabove Proposition. The general case is more mysterious. We point out that (4 . comes down to the following claim: Let f : ˜ X ˇ B → X ˇ G × ˇ T /W ˇ T denote the canonicalmorphism to the fiber product, then we would like to have Rf ∗ O ˜ X ˇ B = O Y ˇ G . To give at least a partial motivation for this expectation let us point out that weexpect the A ˇ G -module Γ( X ˇ G , ˜ M ˇ G ) to have good homological properties. In factthe following conjecture is motivated by the Taylor-Wiles patching construction. Conjecture 4.18.
The A ˇ G -module Γ( X ˇ G , ˜ M ˇ G ) is a maximal Cohen-Macaulaymodule.Remark . One deduces easily from Proposition 4.15 that ˜ M G can not be flatas an O X ˇ G -module. Our family ˜ M G agrees up to twist with | det | − ( n − / (andup to some flat base changes) with the family proposed by Emerton and Helm[11] constructed by Helm in [16]. Hence it should be expected that ˜ M G satisfieslocal-global compatibility with the cohomology of locally symmetric spaces in acertain sense. A precise formulation would include the (derived) base change to aglobal Galois deformation ring. At least for generic representations there should beno obstruction for this base change to sit in a single cohomological degree. Thismotivates the following observation. Corollary 4.20.
Let x = ( ϕ, N ) ∈ X regˇ G such that the ˇ G -conjugacy class of ( ϕ, N ) is a generic L -parameter. Then ˜ M G is locally free (as an O X ˇ G -module) in a neigh-borhood of x .Proof. This follows from Remark 2.9 and the identification of ˜ M G | X regˇ G above. (cid:3) The main conjecture in the regular case.
After restricting to the regularcase we give a candidate for the functor R ψG in Conjecture 3.2, as well as functors R ψ M M for all (standard) Levi subgroups, and prove compatibility with parabolicinduction. As in the case of GL n the choice of ( B , ψ ) is unique up to conjugation,we will always omit the superscript ψ from the notation. By abuse of notation wewill also use the symbols ι GP ( − ) and ι GP ( − ) to denote the functors on Hecke modulescorresponding to parabolic induction (3 . .For a standard Levi subgroup Q si =1 GL r i = M ⊂ G = GL n we write M M for thetensor product of the pullbacks of the M GL ri ( F ) on X GL ri to X ˇ M = Q si =1 X GL ri .This is a sheaf of O [ X ˇ M / ˇ M ] ⊗ Z M H M -modules. We define the functor(4.11) R M : D + ( H M -mod ) −→ D +QCoh ([ X ˇ M / ˇ M ]) π • t π • ⊗ L H M M M . The derived tensor product in the formula can be defined for bounded above objects π • ∈ D + ( H M -mod ) using finite projective resolutions (recall that H M has finite global dimension). In general an object π • ∈ D + ( H M -mod ) can be written as thedirect limit of its truncations lim −→ τ Using the above identifications we can define this as the composition(4.14) Lβ ∗ M M = ( Lβ ∗ ◦ Rβ , ∗ ◦ Rβ ∗ )( O [ X regˇ BM / ˇ B M ] ) −→ Rβ ∗ Lα ∗ ( O [ X regˇ BM / ˇ B M ] ) ∼ = −→ Lα ∗ Rβ M , ∗ ( O [ X regˇ BM / ˇ B M ] ) = Lα ∗ M M , where the first morphism is given by adjunction and the second morphism is givenby the base change morphism in the cartesian square in (4 . . A priori this is onlya morphism of O [ X regˇ P / ˇ P ] -modules. Lemma 4.21. The morphism (4 . is a morphism of H M -modules.Proof. We prove the claim after pulling back to ˜ X regˇ P in (4 . .We write ˜ α , ˜ β etc. for the corresponding morphisms of schemes. As all the maps ˜ β (with various subscripts) are affine, all but the first object in (4 . are concentratedin degree 0. Moreover, all schemes are reduced, and hence it is enough to prove theclaim after restricting to the dense open subscheme where ϕ is regular semi-simple.We denote these open subschemes by ˜ X reg - ssˇ B M etc.. Consider the diagram ˜ X reg - ssˇ B M / / ˜ α (cid:15) (cid:15) X reg - ssˇ M × ˇ T /W M ˇ T / / ˇ M reg - ss2 × ˇ T/W M ˇ T / / ( ∗ ) (cid:15) (cid:15) ˇ T ˜ X reg - ssˇ B M / / X reg - ssˇ M × ˇ T /W M ˇ T / / ˇ M reg - ss1 × ˇ T/W M ˇ T / / ˇ T . Here, the vertical arrow ( ∗ ) on the right hand side is induced by the identification ˇ M reg - ss i × ˇ T /W Mi ˇ T ∼ = (cid:8) ( ϕ, g ˇ B M i ) ∈ ˇ M reg - ss i × ˇ M i / ˇ B M i | ϕ ∈ g − ˇ B M i g (cid:9) . By definition the H M -module structures on source and target of ˜ β ∗ ˜ α ∗ O ˜ X reg - ssˇ BM ∼ = ˜ α ∗ ˜ β M , ∗ O ˜ X reg - ssˇ BM are induced by two (a priori maybe different) H M -module structures of the struc-ture sheaves O ˜ X reg - ssˇ BM և O X reg - ssˇ M × ˇ T/WM ˇ T which in turn are given by the pullback of an H M -action on A ˇ T . These H M -actionsare given by- the H M action on A ˇ T given by A ˇ T ∼ = H T e M , st ,- the restriction of the H M action on A ˇ T given by A ˇ T ∼ = H T e M , st .By Lemma 4.10 (ii) these actions coincide. (cid:3) We obtain the following first step towards Conjecture 3.2. Theorem 4.22. For each parabolic B ⊂ P ⊂ G with Levi M the restriction of (4 . to the regular locus is a Z M -linear functor R reg M : D + ( H M - mod) −→ D +QCoh ([ X regˇ M / ˇ M ]) . Moreover, for two parabolic subgroups B ⊂ P ⊂ P the natural transformation ξ M P defined in (4 . is a Z M -linear isomorphism.For parabolic subgroups P ⊂ P ⊂ P let M denote the Levi quotient of P and P ⊂ P denote the images of P ⊂ P in M . Then the diagram in Conjecture3.2 (b), applied to P ⊂ P ⊂ M , commutes. Proof. We are left to prove that ξ M P is an isomorphism and that the diagram inConjecture 3.2 (b) commutes. Using truncations and resolutions by free modulesit is enough to prove that ξ M P ( H M ) : M M = t ( H M ⊗ H M H M ) ⊗ H M M M −→ Rβ , ∗ ( Lα ∗ M M ) ∼ = Rβ , ∗ Rβ ∗ O [ X regˇ BM / ˇ B M ] = M M is an isomorphism. However, this is a direct consequence of the construction of ξ M P in (4 . using the base change isomorphism in the cartesian square of (4 . .As ξ M P is the composition of an adjunction morphism and a base change map, thecommutativity of (b) in the conjecture is a consequence of standard compatibilitiesof base change morphisms and adjunctions. (cid:3) Compactly induced representations. We describe the image of the functor R G defined in (4 . on (the I -invariants in) the compactly induced representations c-ind GK σ P . The result parallels, and is motivated by, results of Pyvovarov in [25].Recall from Proposition 2.1 (ii) that the irreducible components of X ˇ G are inbijection with the possible Jordan canonical forms of the nilpotent endomorphism N . For a partition P let Z ˇ G, P denote the irreducible component of X ˇ G, P suchthat the Jordan canonical form of N at the generic point of Z ˇ G, P is given by thepartition P . Then we set X ˇ G, P = [ P(cid:22)P ′ Z ˇ G, P ′ . In particular we have X ˇ G, P min = X ˇ G , and X ˇ G, P max = Z ˇ G, P max is irreducible. Wewill sometimes write X ˇ G, for this irreducible component, as it is the irreduciblecomponent defined by N = 0 . We write η P for the generic point of the irreduciblecomponent Z ˇ G, P . Proposition 4.23. Let P be a partition. Then R G (c-ind GK σ P ) is concentrated indegree and, viewed as a ˇ G -equivariant coherent sheaf on X ˇ G , has support X ˇ G, P .Moreover, R G ((c-ind GK K ) I ) = O X ˇ G, ,R G ((c-ind GK st G ) I ) = O X ˇ G , equipped with their canonical ˇ G -equivariant structures. In particular R G ((c-ind GN ψ ) I [ T, ) = O [ X ˇ G / ˇ G ] . Proof. We will rather calculate the images of H G e P = (c-ind GK Σ P ) I ∼ = (c-ind GK σ ⊕ m P P ) I . Recall that m P min = m P max = 1 . Using Lemma 4.9 we see that the ˇ G -equivariantcoherent sheaf on X ˇ G defined by R G ((c-ind GK Σ P ) I ) is e P H G ⊗ H G ˜ M G .Recall that by definition the sheaf ˜ M G is the sheaf attached to the image of A ˇ G ⊗ Z H G e st −→ Y P ′ LL( ϕ η P′ , N η P′ ) I induced by the surjections H G e st ⊗ Z k ( η P ′ ) → LL( ϕ η P′ , N η P′ ) I of Lemma 4.7.Consequently e P H G ⊗ H G ˜ M G is the sheaf defined by the image of the morphism A ˇ G ⊗ Z e P H G e st −→ Y P ′ e P LL mod ( ϕ η P′ , N η P′ ) I . N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 41 Note that A ˇ G ⊗ Z e P H G e st is a free A ˇ G -module of rank m P , by Lemma 4.13 (iv).To show that the sheaf R G ( H G e P ) has support X ˇ G, P it remains to show that e P LL( ϕ η P′ , N η P′ ) I = 0 ⇐⇒ P (cid:22) P ′ . The left hand side can be identified with Hom H G ( H G e P , LL( ϕ η P′ , N η P′ ) I ) = Hom G (c-ind GK Σ P , LL( ϕ η P′ , N η P′ ))= Hom K ( σ P , LL( ϕ η P′ , N η P′ )) m P . As LL( ϕ η P′ , N η P′ ) is absolutely irreducible and generic [27, Theorem 3.7] impliesthe claim.If P ∈ {P min , P max } , then Σ P = σ P and A ˇ G ⊗ Z e P H G e st ∼ = A ˇ G . In this case theabove discussion shows that R G ((c-ind GK σ P ) I ) is the structure sheaf of the unionof those irreducible components Z ˇ G, P ′ such that Hom K ( σ P , LL( ϕ η P′ , N η P′ )) = 0 .If P = P max this implies P ′ = P as above. On the other hand, if P = P min , then Hom K ( σ P , LL( ϕ η P′ , N η P′ )) = 0 for all P ′ by [24, Theorem 1.3]. (cid:3) Remark . A closer analysis of the proof shows that R G (c-ind GK σ P ) can neverbe (locally) free over its support X ˇ G, P unless m P = 1 . Indeed, generically on Z ˇ G, P the sheaf R G (c-ind GK σ P ) is free of rank , using [27, Theorem 3.7 (ii)]. Onthe other hand, generically on X ˇ G, this sheaf is free of rank m P . Indeed, let L be the algebraic closure of k ( η P max ) . Then LL( ϕ P max , N P max ) ⊗ k ( η P max ) L is anirreducible representation induced from the upper triangular Borel. On the otherhand c-ind GK σ P ⊗ Z K is a direct sum of m P copies of the same irreducible principalseries representation by Corollary 6.1 and Lemma 6.4 of [23].4.7. Proof of the conjecture for GL . We prove Conjecture 3.2 in the twodimensional case. In this subsection we use the notation G = GL ( F ) and ˇ G is thealgebraic group GL over C . In this case the ˇ B -action on Lie ˇ B ∩N GL has two orbitsand hence X ˇ B is a complete intersection and X ˇ B = X ˇ B , see Remark 2.8. Moreover,this remark implies that X ˇ B is reduced, and both of its two irreducible componentsare the closure of an irreducible component of X regˇ B . To simplify notations, we willwrite X = X ˇ G, = Z ˇ G, P max ⊂ X ˇ G for the component given by N = 0 and X = X ˇ G .Moreover, we sometimes write X = Z ˇ G, P min ⊂ X for the component on which N is generically non-trivial.Using the notations from Remark 4.17 we obtain an identification M G = R ˜ β ∗ O ˜ X ˇ B once we prove that Rf ∗ O X ˇ B = O Y ˇ G . As above this is clear over the open subset X regˇ G ⊂ X = X ˇ G . On the other hand the closed complement of X regˇ G has the openneighborhood X \ X which is an open subset of G . The claim now follows from thewell known fact that h : g GL = { ( ϕ, gB ) ∈ ˇ G × ˇ G/ ˇ B | ϕ ∈ gBg − } −→ ˇ G = GL has vanishing higher direct images, and its global sections are given by Γ( g GL , O g GL ) = Γ(GL × ˇ T /W ˇ T , O GL × ˇ T/W ˇ T ) . As a consequence we still can use (4 . to define a natural transformation(4.15) ξ GB : R G ◦ ι GB −→ ( Rβ ∗ Lα ∗ ) ◦ R T , where α : [ X ˇ B / ˇ B ] → [ X ˇ T / ˇ T ] and β : [ X ˇ B / ˇ B ] → [ X ˇ G / ˇ G ] are the canonical mor-phisms. The same computation as in the proof of Theorem 4.22 again shows thatthis natural transformation is a Z -linear isomorphism. Theorem 4.25. Let G = GL ( F ) and T ⊂ B ⊂ G denote the standard maximaltorus respectively the standard Borel. The functors R G and R T defined by (4 . are fully faithful and the natural transformation ξ GB defined by (4 . is a Z -linearisomorphism. Moreover, R G ((c-ind GN ψ ) I [ T, ) ∼ = O [ X ˇ G / ˇ G ] for a choice of a generic character ψ : N → C × of the unipotent radical N of B . By the above discussion, it remains to show that R G is fully faithful. Let us write f ∈ Z for the element corresponding to the characteristic polynomials of the form ( T − λ )( T − qλ ) for some indeterminate λ . Then the morphism O [ X/ ˇ G ] · f / / O [ X/ ˇ G ] factors through O [ X/ ˇ G ] ։ O [ X / ˇ G ] and yields a morphism(4.16) O [ X / ˇ G ] / / O [ X/ ˇ G ] with image f O [ X/ ˇ G ] and cokernel O [ X / ˇ G ] . Proposition 4.26. Let F , G ∈ {O [ X/ ˇ G ] , O [ X / ˇ G ] } . Then Ext i [ X/ ˇ G ] ( F , G ) = ( Z , i = 00 , i = 0 . More precisely, a Z -basis of Hom = Ext is given by the identity if F = G .If F = O [ X/ ˇ G ] and G = O [ X / ˇ G ] , then a Z -basis is given by the canonical projection,and if F = O [ X / ˇ G ] and G = O [ X/ ˇ G ] , a Z -basis is given by the morphism (4 . .Proof. We can easily reduce to the case C algebraically closed.Consider the canonical projection f : [ X/ ˇ G ] −→ [ ∗ / ˇ G ] = B ˇ G. We need to compute H i (B ˇ G, Rf ∗ R H om ( F , G )) . As ˇ G is reductive and the base field C has characteristic the category of ˇ G -representations is semi-simple and hencethis vector space is given by H i ( Rf ∗ R H om ( F , G )) ˇ G , compare also [9, Lemma 2.4.1].Here we write H i ( Rf ∗ R H om ( F , G )) for the i -th cohomology sheaf of the complex Rf ∗ R H om ( F , G ) , which is a sheaf on B ˇ G , and hence a ˇ G -representation.Let us write ˜ F and ˜ G for the pullbacks of F and G to X . Then, by definition,giving the quasi-coherent sheaf H i ( Rf ∗ R H om ( F , G )) on B ˇ G is the same as givinga ˇ G -equivariant structure on H i ( R H om ( ˜ F , ˜ G )) . We conclude that Ext i [ X/ ˇ G ] ( F , G ) = (cid:0) Ext iX ( ˜ F , ˜ G ) (cid:1) ˇ G , for the canonical ˇ G -representation on Ext iX ( ˜ F , ˜ G ) induced by the ˇ G -equivariantstructures on ˜ F and ˜ G .If F = O [ X/ ˇ G ] , then Ext iX ( ˜ F , ˜ G ) = ( Γ( X, ˜ G ) , i = 00 , i = 0 and one easily computes Γ( X, ˜ G ) ˇ G ∼ = Z in both cases. Moreover, a Z -basis is easilyidentified with the identity, respectively the canonical projection, as claimed. N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 43 Now assume that F = O [ X / ˇ G ] . We compute Ext iX ( ˜ F , ˜ G ) = Γ( X, E xt i O X ( ˜ F , ˜ G )) . If i = 0 , the sheaf E xt i O X ( ˜ F , ˜ G ) clearly is supported on the intersection X ∩ X .We first show that it is a locally free sheaf on X ∩ X (equipped with the reducedscheme structure). Let X reg-ss ⊂ X denote the Zariski open subset of ( ϕ, N ) with ϕ regular semi-simple. Then X ∩ X ⊂ X reg-ss .Moreover, let X ′ → X reg-ss denote the scheme parametrizing a ϕ -stable subspace.This is an étale Galois cover of degree two and the filtration by the universal ϕ -stable subspace has a canonical ϕ -stable splitting. Let us write V and V for theseeigenspaces and let Y → ˜ X ′ denote the ˇ T -torsor trivializing V and V . Moreover,let Z = { ( λ , λ , a, b ) ∈ ˇ T × A | a ( λ − qλ ) = 0 = b ( λ − qλ ) , λ = λ } equipped with the ˇ T = Spec C [ s ± , s ± ] -action that is trivial on ˇ T and via multipli-cation with the character α : ( s , s ) s s − on a , and via α − on b . We considerthe diagram Y β { { ①①①①①①①①① γ (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ X reg-ss Z, where β is the canonical projection, which is ˇ G -equivariant, and γ is the ˇ T -equivariant ˇ G -torsor that is given by writing the matrices of ϕ and N over Y as Mat( ϕ ) = (cid:18) λ λ (cid:19) and Mat( N ) = (cid:18) ab (cid:19) in the chosen basis of V and V .Let x = ( ϕ , ∈ X ∩ X be a C -valued point. Without loss of generalitywe may assume ϕ = diag( λ , qλ ) . Let y ∈ Y be a pre-image of x and let z denote its image in Z , such that z = ( λ , qλ , , . Consider the closed subscheme Z = V ( a, b ) ⊂ Z and write F Z = O Z and G Z = ( O Z , if ˜ G = O X O Z , if ˜ G = O X . Then E xt i O X ( ˜ F , ˜ G ) is locally free on X ∩ X if and only if E xt i O Z ( F Z , G Z ) is locallyfree on Z . Let S = ˆ O Z,z ∼ = C [[ t , t , a ]] / (( t − t ) a ) be the complete local ring at z with λ = λ + t and λ = q ( λ + t ) , and consider the ˇ T -equivariant resolutionof ˆ F Z,z = S/ ( a ) given by · · · → S (2) · a −→ S (1) · ( t − t ) −−−−−→ S (1) · a −→ S. Here S ( m ) is the free S -module of rank with the ˇ T -action twisted by the multi-plication with α m . It follows that Ext iS ( S/ ( a ) , S ) = ( S/ ( a ) , i = 00 , i ≥ , and Ext iA ( S/ ( a ) , S/ ( a )) = S/ ( a ) , i = 00 , i odd ( S/ ( t − t , a ))( − i/ , i ≥ even . In particular E xt i O Z ( F Z , O Z ) vanishes for i = 0 , and E xt i O Z ( F Z , O Z ) vanishes forodd i and is locally free of rank 1 over Z for non-zero even i . We deduce E xt i O X ( ˜ F , O X ) = 0 for i = 0 . Moreover, it follows that E xt i O X ( ˜ F , O X ) vanishes for odd i and is locally free ofrank on X ∩ X for non-zero even i . In particular a ˇ G -invariant global section h ∈ Ext iX ( ˜ F , O X ) = Γ( X, E xt i O X ( ˜ F , O X )) vanishes if h ( x ′ ) ∈ E xt i O X ( ˜ F , O X ) ⊗ k ( x ′ ) for all x ′ ∈ X ∩ X . Hence we have to show h ( x ′ ) = 0 for all x ′ ∈ X ∩ X foreven i = 0 . Again it is enough to check this for our choice x = ( ϕ , . Then ˇ T = Stab GL ( ϕ ) acts on the fiber E xt i O X ( ˜ F , O X ) ⊗ k ( x ) , and h ( x ) is ˇ T -invariant.By the above diagram the ˇ T -action on this fiber is the same as the ˇ T -action on E xt i O Z ( F Z , O Z ) ⊗ k ( z ) = ( , i odd C ( − i/ , i ≥ even . Obviously, for i = 0 , there are no non-trivial ˇ T -invariants.It remains to show that Hom [ X/ ˇ G ] ( O [ X / ˇ G ] , G ) ∼ = Z , and to identify the basisvector. If G = O [ X / ˇ G ] this is clear, and a Z -basis is clearly given by the identity.If G = O [ X/ ˇ G ] , one computes that the pull back of the morphism (4 . to Y specializes to the pullback of a basis vector of H om O Z ( O Z , O Z ) ⊗ k ( γ ( y )) at everypoint of y ∈ Y . The claim easily follows from this. (cid:3) Corollary 4.27. Let D , D ∈ {H G e K , H G e st } . The functor R G induces isomor-phisms Ext i H G ( D , D ) −→ Ext i [ X/ ˇ G ] ( R G ( D ) , R G ( D )) . Proof. Note that H G e K and H G e st are projective and Hom H G ( D , D ) ∼ = Z . ByProposition 4.26 the claim is true for i = 0 and we are left to show that in degree the canonical morphism identifies basis vectors. This is clear if D = D . Let uswrite γ : H G e K → H G e st and γ : H G e st → H G e K for choices of basis vectors andlet f ∈ Z as defined before Proposition 4.26. Then, up to scalars in Z × , we have(4.17) γ ◦ γ = f · id H G e K and γ ◦ γ = f · id H G e st . Writing δ i = R G ( γ i ) one checks that the equalities δ ◦ δ = f · id O [ X / ˇ G ] and δ ◦ δ = f · id O [ X/ ˇ G ] enforce that δ ∈ Hom [ X/ ˇ G ] ( O [ X / ˇ G ] , O [ X/ ˇ G ] ) and δ ∈ Hom [ X/ ˇ G ] ( O [ X/ ˇ G ] , O [ X / ˇ G ] ) are basis vectors. (cid:3) Proof of Theorem 4.25. We show that R G : D b ( H G -mod fg ) −→ D b Coh ([ X ˇ G / ˇ G ]) is fully faithful. The general case then follows from a limit argument as in Remark3.3 (a).By standard arguments the proof boils down to Corollary 4.27: let D • , D • becomplexes in D b ( H G -mod fg ) . We may choose representatives of D • i consisting ofbounded complexes whose entries are direct sums of copies of H G e K and H G e st . N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 45 Assume first D • = H G e K or H G e st concentrated in degree . We prove the claimby induction on the length of D • . By Corollary 4.27 the claim is true if D • haslength , i.e. if D • is concentrated in a single degree. Assume the claim is true forall complexes of length ≤ m and let D • be a complex in degrees [ r, r + m + 1] forsome r ∈ Z . Then D • can be identified with the mapping cone of a morphism ofcomplexes D r [ − r ] −→ ˜ D • with ˜ D • concentrated in degree [ r, r + m ] . The claim follows from the inductionhypothesis, Corollary 4.27 and the long exact cohomology sequence.The general case follows by a similar induction on the length of D • . (cid:3) Calculation of examples. We finish by computing the image of some specialrepresentations under the functor R G defined in (4 . . In particular we are in thesituation G = GL n ( F ) and ˇ G is the algebraic group GL n over C . For simplicitywe assume that C is algebraically closed. We fix the choice of the diagonal torus T and the upper triangular Borel subgroup B .Let x = ( ϕ, N ) ∈ X ˇ G . In the examples calculated in this section we will assumethat ϕ is regular semi-simple. As in Remark 3.3 (b) we write X ◦ ˇ G, [ ϕ,N ] = ˇ G · x for the ˇ G -orbit of ( ϕ, N ) and X ˇ G, [ ϕ,N ] for its closure. Theorem 4.28. Let ( ϕ, N ) ∈ X ˇ G ( C ) and assume that ϕ is regular semi-simple.Then R G (LL mod ( ϕ, N )) = O [ X ˇ G, [ ϕ,N ] / ˇ G ] . To prove this, we will use compatibility with parabolic induction. Hence the mainstep will be to calculate the image of the generalized Steinberg representations.Let χ : Z → C be the character defined by the characteristic polynomial of ϕ .We write ˆ Z χ for the completion of Z with respect to the kernel m χ of χ and ˆ H G,χ = H G ⊗ Z ˆ Z χ for the m χ -adic completion of H G . Similarly, if M ⊂ G is a Levi subgroup, we write ˆ H M,χ for the corresponding completion of H M .Assume that ϕ = diag( ϕ , . . . , ϕ n ) . For w ∈ W = S n we write wϕ for thediagonal matrix diag( ϕ w (1) , . . . , ϕ w ( n ) ) . We use the notation δ w to denote the H T -module defined by the unramified character unr wϕ (i.e. the residue field at the point wϕ ∈ Spec H T ), and ˆ δ w to denote the completion of H T at the point wϕ ∈ Spec H T .Then δ w and ˆ δ w are ˆ H T,χ -modules.We recall intertwining operators for parabolic induction: Let P , P ′ ⊂ G be para-bolic subgroups (containing T ) with Levi subgroups M and M ′ and let w ∈ W suchthat M ′ = wM w − . Let π and π ′ be smooth representations of M respectively M ′ and let f : π w → π ′ be a morphism of M ′ -representations. Then there is acanonical morphism of G -representations F ( w, f ) = F G ( w, f ) : ι GP π −→ ι GP ′ π ′ associated to f (and similarly for ι GP and ι GP ′ ). Moreover, this construction extendsto (morphisms of) complexes of M - respectively M ′ -representations. We also notethat the formation of these intertwining operators is transitive in the followingsense: Let P , P ′ ⊂ P ⊂ G be parabolic subgroups. Let M , M ′ and M denote the corresponding Levi quotients and let P ,M be the image of P in M (and similarly P ′ ,M ). Let w ∈ W M ⊂ W be a Weyl group element such that wM w − = M ′ (as subgroups of G , and hence also as subgroups of M ). Moreover, let π be arepresentation of M and π ′ be a representation of M ′ and f : π w → π ′ be amorphism of M ′ -representations. Then, under the canonical identifications ι GP ι MP ,M ( π ) = ι GP π and ι GP ι MP ′ ,M ( π ′ ) = ι GP ′ π ′ , the morphism ι GP ( F M ( w, f )) is identified with F G ( w, f ) .Now fix λ ∈ C × and let ϕ = diag( λ, q − λ, . . . , q − ( n − λ ) . For w, w ′ ∈ W theidentity of δ w ′ respectively ˆ δ w ′ induces intertwining operators(4.18) f ( w, w ′ ) : ι GB ( δ w ) −→ ι GB ( δ w ′ )ˆ f ( w, w ′ ) : ι GB (ˆ δ w ) −→ ˆ ι GB (ˆ δ w ′ ) . Note that these morphisms are isomorphisms (with inverse f ( w ′ , w ) respectively ˆ f ( w ′ , w ) ) if and only if for each i the entries q − i λ and q − ( i +1) λ appear in the sameorder in wϕ and w ′ ϕ . Moreover,(4.19) Hom ˆ Z χ [ G ] ( ι GB ˆ δ w , ι GB ˆ δ w ′ ) = ˆ Z χ ˆ f ( w, w ′ ) is a free ˆ Z χ -module of rank . We define the (universal) deformation of the gener-alized Steinberg representation ˆSt( λ, r ) = ι GB (cid:0) δ − / B ⊗ d unr λ | − | ( n − / (cid:1)(cid:14) X B ( P ⊆ G ι GP (cid:0) δ − / P ⊗ d unr λ | − | ( n − / (cid:1) . Here we write d unr λ for the universal (unramified) deformation of the character unr λ and denote the target of d unr λ by C [[ t ]] . Then d unr λ ⊗ C [[ t ]] C [[ t ]] / ( t ) = unr λ , ˆSt( λ, r ) ⊗ C [[ t ]] C [[ t ]] / ( t ) = St( λ, r ) . Note that by definition ˆSt( λ, n ) is a quotient of ι GB ˆ δ w , where w ∈ W is the longestelement.By abuse of notation we will also write St( λ, n ) and ˆSt( λ, n ) for the H G - re-spectively ˆ H G,χ -module given by the I -invariants in the respective representations.Similarly, we will continue to write ι GB δ w etc. for the Hecke modules defined by theserepresentations. In the following we will only work with Hecke modules, hence noconfusion should arise.We construct a projective resolution ˆ C • n,λ of the ˆ H G,χ -module ˆSt( λ, n ) concen-trated in (cohomological) degrees [ − ( n − , such that all objects in the complexare direct sums of induced representations ι GB ˆ δ w and the differntials are given bycombinations of the intertwining morphisms (4 . . We construct the complex byinduction.If n = 2 , then ϕ = diag( λ, q − λ ) and we consider the complex ˆ C • ,λ : ˆ C − ,λ = ι GB (ˆ δ ) ˆ f (1 ,s ) / / ˆ C ,λ = ι GB (ˆ δ s ) , where s ∈ S is the unique non trivial element. It can easily be checked that themorphism ˆ f (1 , s ) is injective and that its cokernel is ˆSt( λ, .Assume we have constructed ˆ C • n − ,λ . For i = 1 , consider the upper triangularblock parabolic subgroup P i ⊂ G with Levi subgroup M i such that M has block N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 47 sizes ( n − , and M has block sizes (1 , n − . We consider D • = ˆ C • n − ,λ b ⊗ d unr q − ( n − λ as a complex of ˆ H M ,χ -modules and D • = d unr q − ( n − λ b ⊗ ˆ C • n − ,λ as a complex of ˆ H M ,χ -modules. Let σ ∈ S n be the cycle (12 . . . n ) . Then M and M satisfy σM σ − = M and the identity ( D • ) σ → D • , as a morphism ofcomplexes of ˆ H M ,χ -modules, induces a morphism of complexes ι GP D • −→ ι GP D • . We define ˆ C • n,λ as the mapping cone of this complex. Then ˆ C • n,λ obviously is a com-plex in degree [ − ( n − , whose entries are (by transitivity of parabolic induction)direct sums of ι GB (ˆ δ w ) for some w ∈ W (each isomorphism class appearing exactlyonce) and the differentials are given by intertwining operators (by transitivity ofintertwining operators). Lemma 4.29. The complex ˆ C • n,λ is exact in negative degrees and H ( ˆ C • n,λ ) ∼ = ˆSt( λ, n ) . Proof. We proceed by induction. If n = 2 this was already remarked above. Assumethat ˆ C • n − ,λ is quasi-isomorphic to ˆSt( λ, n − . Then ˆ C • n,λ is quasi-isomorphic tothe complex ι GP (cid:0) ˆSt( λ, n − b ⊗ d unr q − ( n − λ (cid:1) −→ ι GP (cid:0)d unr q − ( n − λ b ⊗ ˆSt( λ, n − (cid:1) in degrees − and , where the morphism is given by the obvious intertwining map.One can easily check that this morphism is injective and its cokernel is ˆSt( λ, n ) . (cid:3) Similarly to the definition of ˆ H G,χ we define m χ -adic completions on the side ofstacks of L-parameters: let ˆ X ˇ G,χ denote the completion of X ˇ G along the pre-imageof χ ∈ Spec Z = ˇ T /W under the canonical morphism X ˇ G → ˇ T /W . This formalscheme is still equipped with an action of ˇ G and we can form the stack quotient [ ˆ X ˇ G,χ / ˇ G ] . Similarly we write ˆ X ˇ P ,χ and ˆ X ˇ M,χ for the corresponding completions of X ˇ P and X ˇ M . The functor R G defined in (4 . naturally extends to a functor ˆ R G,χ : D + ( ˆ H G,χ -mod ) −→ D +QCoh ([ ˆ X ˇ G,χ / ˇ G ]) . As a consequence of Theorem 4.22 the functor ˆ R G,χ also satisfies compatibility withparabolic induction similarly to Conjecture 3.2 (ii), but for the induced morphismbetween the formal completions of the stacks involved.Let us build a more explicit model of these stacks. We consider the closed formalsubscheme(4.20) ˆ Y = Spf (cid:0) C [[ t , . . . , t n ]][ u , . . . , u n − ] / (( t i +1 − t i ) u i ) (cid:1) ⊂ ˆ X ˇ G,χ , where C [[ t , . . . , t n ]][ u , . . . , u n − ] / (( t i +1 − t i ) u i ) is equipped with the ( t , . . . , t n ) -adic topology. The embedding into ˆ X ˇ G,χ is defined by the ( ϕ, N ) -module ϕ ˆ Y = diag( λ + t , q − ( λ + t ) , . . . , q − ( n − ( λ + t n )) N ˆ Y ( e i ) = ( u i e i +1 , i < n − , i = n − over ˆ Y . This formal scheme comes equipped with a canonical ˇ T -action (which istrivial on the t i and via the adjoint action on the u i ) such that [ ˆ Y / ˇ T ] = [ ˆ X ˇ G,χ / ˇ G ] .For w ∈ W we define a closed ˇ T -equivariant formal subscheme ˆ Y ( w ) by addingthe equation u i = 0 if q − ( i − λ precedes q − i λ in wϕ . In particular ˆ Y ( w ) = ˆ Y if w ∈ W is the longest element. We denote by ˆ X ˇ G,χ ( w ) the corresponding ˇ G -equivariant closed formal subscheme of ˆ X ˇ G,χ . Lemma 4.30. There is an isomorphism ˆ R G,χ ( ι GB ˆ δ w ) ∼ = O ˆ X ˇ G,χ ( w ) , where we view ˆ R G,χ ( ι GB ˆ δ w ) as a ˇ G -equivariant sheaf on ˆ X ˇ G,χ .Proof. This is a straight forward calculation using the compatibility of ˆ R G,χ withparabolic induction. (cid:3) The lemma identifies the images of parabolically induced representations under ˆ R G,χ . Next we identify the images of intertwining operators. For w, w ′ ∈ W thereis a canonical ˇ T -equivariant morphism ˆ g ( w, w ′ ) : O ˆ Y ( w ) −→ O ˆ Y ( w ′ ) defined as follows: let I w = { i = 1 , . . . , n − | q − ( i − λ precedes q − i λ in wϕ } , i.e. ˆ Y ( w ) = Spf C [[ t , . . . , t n ]][ u , . . . , u n − ] / ( u i , i ∈ I w , ( t i +1 − t i ) u i , i / ∈ I w ) and let us write ˆ Y ( w, w ′ ) = Spf C [[ t , . . . , t n ]][ u , . . . , u n − ] / ( u i , i ∈ I w ∩ I w ′ , ( t i +1 − t i ) u i , i / ∈ I w ∩ I w ′ ) for the moment. Then, similarly to (4 . , multiplication by Q i ∈ I w \ I w ′ ( t i +1 − t i ) induces a morphism O ˆ Y ( w ) → O ˆ Y ( w,w ′ ) and we define ˆ g ( w, w ′ ) to be its compositionwith the canonical projection to O ˆ Y ( w ′ ) . Lemma 4.31. For w, w ′ ∈ W the ˆ Z χ -module Hom [ ˆ X ˇ G,χ / ˇ G ] ( O [ ˆ X ˇ G,χ ( w ) / ˇ G ] , O [ ˆ X ˇ G,χ ( w ′ ) / ˇ G ] ) = Hom [ ˆ Y / ˇ T ] ( O [ ˆ Y ( w ) / ˇ T ] , O [ ˆ Y ( w ′ ) / ˇ T ] ) is free of rank one with basis ˆ g ( w, w ′ ) .Proof. This is a straight forward computation. (cid:3) By the following theorem the images of the intertwining operators ˆ R G,χ ( ˆ f ( w, w ′ )) can be identified (up to isomorphism) with the morphisms ˆ g ( w, w ′ ) just constructed. Theorem 4.32. Let ϕ = diag( λ, q − λ, . . . , q − ( n − λ ) ∈ ˇ T ( C ) and χ : Z → C thecharacter defined by the image of ϕ in ˇ T /W . The set of functors ˆ R M,χ : D + ( ˆ H M,χ - mod) −→ D +QCoh ([ ˆ X ˇ M,χ / ˇ M ]) for standard Levi subgroups M ⊂ G , is uniquely determined (up to isomorphism)by requiring that they are ˆ Z M,χ -linear, compatible with parabolic induction, and that ˆ R T,χ is induced by the identification ˆ H T,χ - mod ∼ = −→ QCoh( ˆ X ˇ T ,χ ) . More precisely, let ˆ R ′ G,χ be any functor satisfying these conditions. Then for each w ∈ W , there are isomorphisms α w : ˆ R ′ G,χ ( ι GB ˆ δ w ) ∼ = −→ O ˆ X ˇ G,χ ( w ) N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 49 such that for w, w ′ ∈ W the diagram (4.21) ˆ R ′ G,χ ( ι GB ˆ δ w ) α w / / ˆ R ′ G,χ ( ˆ f ( w,w ′ )) (cid:15) (cid:15) O ˆ X ˇ G,χ ( w )ˆ g ( w,w ′ ) (cid:15) (cid:15) ˆ R ′ G,χ ( ι GB ˆ δ w ′ ) α w ′ / / O ˆ X ˇ G,χ ( w ′ ) commutes.Remark . (a) Note that we do not need to add the requirement ˆ R G,χ ((c-ind GN ψ ) I [ T, ⊗ Z ˆ Z χ ) ∼ = O [ ˆ X ˇ G,χ / ˇ G ] which also would be a consequence of the requirements in Conjecture 3.2. In thesituation considered here, there is an isomorphism (c-ind GN ψ ) [ T, ⊗ Z ˆ Z χ ∼ = ι GB ˆ δ w and hence the above isomorphism is automatic.(b) It seems possible to compute that Ext i [ ˆ Y / ˇ T ] ( O [ ˆ Y ( w ) / ˇ T ] , O [ ˆ Y ( w ′ ) / ˇ T ] ) = 0 for w, w ′ ∈ W and i = 0 by a similar explicit computation as in Proposition 4.26.This would imply the conjectured fully faithfulness of ˆ R G,χ . Proof. Let us first justify that the second assertion implies the first. Note that ˆ H M,χ = ˆ H M,χ ⊗ ˆ H T,χ ˆ H T,χ = ˆ H M,χ ⊗ ˆ H T,χ (cid:0) M w ∈ W M ˆ δ w (cid:1) = M w ∈ W M ι MB M ˆ δ w . Using free resolutions of bounded above objects in D + ( ˆ H M,χ -mod ) it is henceenough control the images of parabolically induced representations and the imagesof the intertwining operators. Then a limit argument deals with the general case.Given ˆ R ′ G,χ as in the formulation of the theorem, compatibility with parabolicinduction forces the existence of isomorphisms α w . Note that α w is unique up to aunit in ˆ Z χ . We claim that we can choose the isomorphisms such that the diagrams (4 . are commutative. In order to do so, we proceed by induction. By assumptionthe claim is true for n = 1 . We also make n = 2 explicit. In this case we can identify ι GB ˆ δ = ˆ H G,χ e K and ι GB ˆ δ s = ˆ H G,χ e st . One calculates that the intertwining operators ˆ f (1 , s ) and ˆ f ( s, are identified witha ˆ Z χ -basis of Hom ˆ H G,χ ( ˆ H G,χ e K , ˆ H G,χ e st ) resp. Hom ˆ H G,χ ( ˆ H G,χ e st , ˆ H G,χ e K ) . Moreover, the compositions ˆ f (1 , s ) ◦ ˆ f ( s, and ˆ f ( s, ◦ ˆ f (1 , s ) are the multiplica-tions with f ∈ ˆ Z × χ , with f as defined just before Proposition 4.26. The calculationin the rank case, Proposition 4.26, yields the claim.Assume now that the claim is true for n − and view S n − as the subgroup of W = S n permuting the elements , . . . , n − . Recall the parabolic subgroups P and P from the inductive construction of the complex ˆ C • n,λ . Using parabolic induction ι GP and the induction hypothesis we may assume that we have constructed α w forall w ∈ S n − ⊂ W such that the diagram (4 . commutes for all w, w ′ ∈ S n − .Let σ = (12 . . . n ) as above. We first show that we can choose α σwσ − : ˆ R ′ G,χ ( ι GB ˆ δ σwσ − ) ∼ = −→ O ˆ Y ( σwσ − ) such that (4 . commutes for the pairs w, σwσ − and σwσ − , w . Let τ i,i +1 denotethe transposition of i and i + 1 . Inductively we define w = τ n,n − wτ n,n − and w i = τ n − i,n − i − w i − τ n − i,n − i − . Then the composition of intertwining operators ι GB ˆ δ w −→ ι GB ˆ δ w −→ · · · −→ ι GB ˆ δ w n − = ι GB ˆ δ σwσ − is identified with β ˆ f ( w, σwσ − ) for some β ∈ ˆ Z × χ . Similarly, the composition ι GB ˆ δ σwσ − −→ ι GB ˆ δ w n − −→ · · · −→ ι GB ˆ δ w = ι GB ˆ δ w is identified with β ˆ f ( σwσ − , w ) for the same unit β . In this composition all theintertwining maps are isomorphisms, except for the morphisms ι GB ˆ δ w i −→ ι GB ˆ δ w i +1 and ι GB ˆ δ w i +1 −→ ι GB ˆ δ w i where the position of n and n − in ( w i (1) , . . . , w i ( n )) and ( w i +1 (1) , . . . , w i +1 ( n )) is interchanged. By the computation in the two dimensional case and compatibilitywith parabolic induction this intertwining morphism is given by the multiplicationwith β ′ ( t n − t n − ) for some unit β ′ ∈ ˆ Z × χ respectively by canonical projectionmultiplied with β ′ . Modifying α σwσ − by ( ββ ′ ) − we deduce the commutativity ofthe diagrams (4 . for the pairs w, σwσ − and σwσ − , w .Now consider the general case. Note that for any w ∈ W there exists ˜ w ∈ S n − such that ι GB ˆ δ w ∼ = −→ ι GB ˆ δ ˜ w or ι GB ˆ δ w ∼ = −→ ι GB ˆ δ σ ˜ wσ − . Hence we can choose α w such that all the diagrams (4 . commute, provided wecan check commutativity of these diagrams for w, w ′ ∈ S n − ∪ σ S n − σ − . If bothelements w, w ′ lie in S n − this follows from the induction hypothesis. Let us checkthe claim for w, w ′′ ∈ S n − and w ′ = σw ′′ σ − (the argument in the other casesbeing similar). By ˆ Z χ -linearity it is enough to check that ˆ R ′ G,χ ( ι GB ˆ δ w ) α w / / ˆ R ′ G,χ ( γ ˆ f ( w,w ′ )) (cid:15) (cid:15) O ˆ X ˇ G,χ ( w ) γ ˆ g ( w,w ′ ) (cid:15) (cid:15) ˆ R ′ G,χ ( ι GB ˆ δ w ′ ) α w ′ / / O ˆ X ˇ G,χ ( w ′ ) commutes for any choice of = γ ∈ ˆ Z χ . In particular we may check it for theelement γ defined by ˆ f ( w ′′ , σw ′′ σ − ) ◦ ˆ f ( w, w ′′ ) = γ ˆ f ( w, σw ′′ σ − ) = γ ˆ f ( w, w ′ ) . This follows from functoriality and the cases already treated above. (cid:3) We now continue to calculate the image ˆ R G,χ ( ˆSt( λ, n )) of the deformed Steinbergrepresentation. Let us write ˆ Y St ∼ = Spf C [[ t ]][ u , . . . , u n − ] ⊂ ˆ Y for the formal subscheme defined by t := t = · · · = t n . We write ˆ X Stˇ G,χ for thecorresponding ˇ G -equivariant scheme.We inductively construct a ˇ T -equivariant resolution ˆ E • n,λ of O ˆ Y St .If n = 2 we set ˆ E • ,λ : ˆ E − ,λ = O ˆ Y (1) · ( t − t ) / / ˆ E ,λ = O ˆ Y ( s ) , where again s ∈ S is the unique non trivial element. It can easily be checked thatthis morphism is injective and its cokernel is O ˆ Y St N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 51 Assume that ˆ E • n − ,λ is constructed, then consider the morphism of complexes ˆ E • n − ,λ [[ t n ]][ u n − ] / ( u n − ) · ( t n − t n − ) −−−−−−−→ ˆ E • n − ,λ [[ t n ]][ u n − ] / (( t n − t n − ) u n − ) on ˆ Y and define ˆ E • n,λ to be its mapping cone. Lemma 4.34. The complex ˆ E • n,λ is exact in negative degrees and H ( ˆ E • n,λ ) = O ˆ Y St . Proof. We proceed by induction. For n = 2 the claim is clear. Assume the claimis true for n − , then the long exact cohomology sequence implies that ˆ E • n,λ isquasi-isomorphic to the complex C [[ t, t n ]][ u , . . . , u n − ] −→ C [[ t, t n ]][ u , . . . , u n − , u n − ] / (( t n − t ) u n − ) sending to ( t n − t ) . The claim follows from this. (cid:3) Let us denote by ˆ E • n,λ the ˇ G -equivariant complex on ˆ X ˇ G,χ corresponding to the ˇ T -equivariant complex ˆ E • n,λ under the identification [ ˆ Y / ˇ T ] = [ ˆ X ˇ G,χ / ˇ G ] . Corollary 4.35. There is an isomorphism of complexes (4.22) ˆ R G,χ ( ˆ C • n,λ ) ∼ = ˆ E • n,λ . Proof. We prove this using the inductive construction of both complexes. The case n = 1 is trivial. Assume now that (4 . is true for n − . Recall the parabolicsubgroups P and P from the inductive construction of ˆ C • n,λ .Let us write G n − = GL n − ( F ) and B n − ⊂ G n − for the upper triangular Borel.Further let ϕ ′ = diag( λ, q − λ, . . . , q − ( n − λ ) . Similarly to the definition of δ w and ˆ δ w using wϕ we define δ ′ w and ˆ δ ′ w using wϕ ′ for w ∈ S n − . Then ι GP (cid:0) ι G n − B n − ˆ δ ′ w b ⊗ d unr q − ( n − λ (cid:1) = ι GB ˆ δ w ι GP (cid:0)d unr q − ( n − b ⊗ ι G n − B n − ˆ δ ′ w λ (cid:1) = ι GB ˆ δ σwσ − , and the intertwining operator between the representations on the right hand sidetranslates to the intertwining operator ˆ f ( w, σwσ − ) under this identification.By the same inductive construction, we assume that each entry of ˆ C • n − ,λ is a directsum of representations ι G n − B n − ˆ δ ′ w for w ∈ S n − . By Theorem 4.32 the morphism ˆ R G,χ ( ι GP D • ) −→ ˆ R G,χ ( ι GP D • ) is (up to a unit) identified with the multiplication by ( t n − t n − ) . The inductiveconstruction of ˆ E • n,λ hence implies the claim. (cid:3) Corollary 4.36. Let λ ∈ C × and let ( ϕ, N ) ∈ X ˇ G ( C ) be the L-parameter definedby ( C n , ϕ, N ) = Sp( λ, n ) . Then R G (St( λ, n )) ∼ = O X ˇ G, [ ϕ,N ] , where St( λ, n ) = LL( ϕ, N ) = LL mod ( ϕ, N ) is the generalized Steinberg representa-tion.Proof. The corollary above implies ˆ R G,χ ( ˆSt( λ, n )) = O ˆ X Stˇ G,χ as ˇ G -equivariant sheaves. Moreover, we have ˆSt( λ, n ) ⊗ LC [[ t ]] C [[ t ]] / ( t ) = ˆSt( λ, n ) ⊗ C [[ t ]] C [[ t ]] / ( t ) = St( λ, n ) , O ˆ X Stˇ G,χ ⊗ LC [[ t ]] C [[ t ]] / ( t ) = O ˆ X Stˇ G,χ ⊗ C [[ t ]] C [[ t ]] / ( t ) = O X ˇ G, [ ϕ,N ] . The center ˆ Z χ acts on ˆSt( λ, n ) and O ˆ X Stˇ G,χ via a surjection ˆ Z χ −→ C [[ t ]] . Choosing a pre-image g of t we obtain isomorphisms ˆSt( λ, n ) ⊗ LC [[ t ]] C [[ t ]] / ( t ) = ˆSt( λ, n ) ⊗ L ˆ Z χ ˆ Z χ / ( g ) O ˆ X Stˇ G,χ ⊗ LC [[ t ]] C [[ t ]] / ( t ) = O ˆ X Stˇ G,χ ⊗ L ˆ Z χ ˆ Z χ / ( g ) . The claim now follows from ˆ Z χ -linearity of ˆ R G,χ . (cid:3) Remark . With some extra effort one can use a similar strategy to compute theimages of LL( ϕ, N ) , where ϕ = diag( λ, q − λ, . . . , q − ( n − λ ) and N is an arbitraryendomorphism such that ( ϕ, N ) ∈ X ˇ G . Recall that LL( ϕ, N ) is the unique simplequotient of LL mod ( ϕ, N ) . One needs to build a complex similar to ˆ C • n,λ which is aresolution of LL( ϕ, N ) . We omit the technical computation, and only describe theresult.Let us choose such y = ( ϕ, N ) ∈ Y ⊂ X ˇ G , where Y ⊂ ˆ Y is the closed subscheme t = · · · = t n = 0 . We denote by L ( y ) the sheaf of ideals defining the closedsubscheme [ { i | u i ( y )=0 } { u i = 0 } ⊂ Y. Obviously this is a ˇ T -equivariant line bundle and we write L ( y ) for the correspond-ing ˇ G -equivariant line bundle on X ˇ G . Let us denote the number of i ∈ { , . . . , n − } such that u i ( y ) = 0 by l y . Then R G (LL( ϕ, N )) = L ( y )[ l y ] is the equivariant line bundle L ( y ) shifted to (cohomological) degree − l y . Proof of Theorem 4.28. We assume that ϕ is an arbitrary regular semi-simple ele-ment and choose a decomposition ( C n , ϕ, N ) = s M i =1 Sp( λ i , r i ) as in subsection 4.1. Then LL mod ( ϕ, N ) = ι GP (cid:0) St( λ , r ) ⊗ · · · ⊗ St( λ s , r s ) (cid:1) = ι GP ′ (cid:0) St( λ s , r s ) ⊗ · · · ⊗ St( λ , r ) (cid:1) with the ordering of (4 . . Here P is a block upper triangular parabolic with Levi M and we set P ′ to be the block upper triangular parabolic with Levi M ′ = w M w − ,where w ∈ W is the longest element. Write ˇ M ′ = GL r s × · · · × GL r and considerthe morphisms α : X ˇ P ′ −→ X ˇ M ′ ,β : ˜ X ˇ P ′ −→ X ˇ G . The choice of ˇ M ′ ֒ → ˇ P ′ defines an embedding ι : X ˇ M ′ ֒ → X ˇ P ′ . We will write ( x s , . . . , x ) ∈ X ˇ M ′ for the point defined by Sp( λ s , r s ) ⊕ · · · ⊕ Sp( λ , r ) , N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 53 and write Z ˇ M ′ ( x , . . . , x s ) for the Zariski-closure of its ˇ M ′ -orbit ˇ M ′ · ( x s , . . . , x ) .Then one easily checks that the choice of ordering of λ , . . . , λ s implies that α − ( Z ˇ M ′ ( x s , . . . , x )) =: Z ˇ P ′ ( x s , . . . , x ) is the Zariski-closure of the ˇ P ′ -orbit of ι ( x s , . . . , x ) . Moreover, the choice of order-ing implies that α is smooth along this pre-image. In particular Lα ∗ O Z ˇ M ′ ( x s ,...,x ) = O Z ˇ P ′ ( x s ,...,x ) . Let Z ˇ G ( x s , . . . , x ) ⊂ ˜ X ˇ P ′ denote the ˇ G -invariant closed subscheme of ˜ X ˇ P ′ cor-responding to the ˇ P ′ -invariant closed subscheme Z ˇ P ′ ( x s , . . . , x ) ⊂ X ˇ P ′ . UsingCorollary 4.36 and compatibility of R G with parabolic induction, we are left toshow that Rβ ∗ ( O Z ˇ G ( x s , . . . , x )) = O X ˇ G, [ ϕ,N ] . This follows, as the construction implies that β maps Z ˇ G ( x s , . . . , x ) isomorphicallyonto the Zariski-closure X ˇ G, [ ϕ,N ] of the ˇ G -orbit ˇ G · ( ϕ, N ) = ˇ G · ι ( x s . . . , x ) . (cid:3) We also remark that Theorem 4.32 is true for all regular semi-simple elements ϕ . Corollary 4.38. Let ϕ ∈ ˇ T ( C ) be regular semi-simple and χ : Z → C the characterdefined by the image of ϕ in ˇ T /W . The set of functors ˆ R M,χ : D + ( ˆ H M,χ - mod) −→ D +QCoh ([ ˆ X ˇ M,χ / ˇ M ]) for standard Levi subgroups M ⊂ G , is uniquely determined (up to isomorphism)by requiring that they are ˆ Z M,χ -linear, compatible with parabolic induction, and that ˆ R T,χ is induced by the identification ˆ H T,χ - mod = QCoh( ˆ X ˇ T ,χ ) . Proof. As in the proof of Theorem 4.32 the images of ˆ R G,χ (ˆ δ w ) are uniquely deter-mined up to isomorphism and it is enough to prove that the same is true for theimages of intertwining operators. Without loss of generality we may assume ϕ = diag( λ , q − λ , . . . , q − ( r − λ , . . . λ s , q − λ s , . . . q − ( r s − λ s ) with q − a λ i = q − b λ j for i = j, a = 0 , . . . , r i − , b = 0 , . . . , r j − , and λ i = q − r j λ j .Let M = GL r ( F ) × · · · × GL r s ( F ) be the block diagonal Levi subgroup withblock sizes ( r , . . . , r s ) and P the corresponding block upper triangular parabolicsubgroup. Further let ϕ i = diag( λ i , q − λ i , . . . , q − ( r i − λ i ) ∈ GL r i ( F ) . For w i ∈ S r i we write ˆ δ ( i ) w i for the universal unramified deformation of the character defined by w i ϕ i . Then, by means of an intertwining operator, every ι GB ˆ δ w is isomorphic to ι GP ( ι MB M (ˆ δ (1) w ⊗ · · · ⊗ ˆ δ ( s ) w s )) for some ( w , . . . , w s ) ∈ S r × · · · × S r s = W M . As in the proof of Theorem 4.32we deduce that, given two functors ˆ R G,χ and ˆ R ′ G,χ satisfying the assumptions, it isenough to show that for all w ∈ W M there are isomorphisms α w : ˆ R G,χ ( ι GB ˆ δ w ) −→ ˆ R ′ G,χ ( ι GB ˆ δ w ) such that the diagrams ˆ R G,χ ( ι GB ˆ δ w ) α w / / ˆ R G,χ ( ˆ f ( w,w ′ )) (cid:15) (cid:15) ˆ R ′ G,χ ( ι GB ˆ δ w ) ˆ R ′ G,χ ( ˆ f ( w,w ′ )) (cid:15) (cid:15) ˆ R G,χ ( ι GB ˆ δ w ′ ) α w ′ / / ˆ R ′ G,χ ( ι GB ˆ δ w ′ ) commute for all w, w ′ ∈ W M . This follows from the statement of Theorem 4.32 andtransitivity of intertwining operators under parabolic induction. (cid:3) We finish by giving more details on the behavior of R G (c-ind GK σ P ) in the three-dimensional case. Example . In the case n = 3 there are three partitions P min , P , P max of n = 3 .We have m P min = m P max = 1 ,m P = 2 , where the multiplicities are defined as in (4 . . The sheaves R G ((c-ind GK σ min ) I ) and R G ((c-ind GK σ max ) I ) are determined in Proposition 4.23. Let us give a closerdescription of F = R G ((c-ind GK σ P ) I ) . As discussed in the Remark 4.24 the generic rank of F on Z ˇ G, P ′ is if P ′ = P min ,it is if P ′ = P and it is if P ′ = P max .We describe the completed stalks ˆ F x as modules over the complete local rings ˆ O X ˇ G ,x for C -valued points x = ( ϕ, N ) ∈ X ˇ G . To simplify the exposition we restrictourselves to regular semi-simple ϕ . Recall that X ˇ G, P = Z ˇ G, P ∪ Z ˇ G, P max is aunion of two irreducible components in this case. Moreover, recall that we write X ˇ G, = Z ˇ G, P max for the irreducible component defined by N = 0 .(a) Assume x ∈ Z ˇ G, P min \ X ˇ G, P , then, ˆ F x = 0 .(b) Assume x ∈ Z ˇ G, P \ Z ˇ G, P max , then ˆ F x ∼ = ˆ O X ˇ G ,x .(c) Assume x ∈ Z ˇ G, P max \ Z ˇ G, P , then ˆ F x ∼ = ˆ O X ˇ G ,x .(d) Assume x ∈ Z ˇ G, P max ∩ Z ˇ G, P . Without loss of generality we may assume ϕ = diag( λ , λ , λ ) . As before we write χ : Z → C for the character defined bythe characteristic polynomial of ϕ . Up to renumbering, we have to distinguish twocases:(d1) λ = q − λ and λ / ∈ { q − λ , qλ } . In this case Z ˇ G, P and Z ˇ G, P max aresmooth at x . Moreover (using the notations introduced above) c-ind GK σ P ⊗ Z ˆ Z χ ∼ = ι GB ˆ δ w ⊕ ι GB ˆ δ w for some w , w ∈ W and (with appropriate numeration) c-ind GK σ P min ⊗ Z ˆ Z χ ∼ = ι GB ˆ δ w = ι GB ˆ δ w ∼ = c-ind GK σ P max ⊗ Z ˆ Z χ . We then can use compatibility of R G with parabolic induction to deducethat ˆ F x ∼ = ˆ O X ˇ G, P ,x ⊕ ˆ O X ˇ G, ,x . (d2) λ = q − λ = q − λ . In this case Z ˇ G, P is no longer smooth at x , buthas a self intersection as can be seen from the description of the completelocal ring: using a local presentation as in (4 . we can compute that thecomplete local ring of ˆ O X ˇ G ,x is smoothly equivalent to C [[ t , t , t , u , u ]] / (( t − t ) u , ( t − t ) u ) . With these coordinates the completion of Z ˇ G, P min at x is given by thevanishing locus V ( t − t , t − t ) and the completion of Z ˇ G, P max is givenby V ( u , u ) . Moreover, both are smooth at x . However, the completionof Z ˇ G, P is given by V ( t − t , u ) ∪ V ( u , t − t ) , i.e. it decomposes into N THE DERIVED CATEGORY OF THE IWAHORI-HECKE ALGEBRA 55 two components, say ˆ Z and ˆ Z . Note that this computation implies that Z ˇ G, P can not be Cohen-Macaulay at x , as it has a self intersection incodimension . We can compute the completions of the compactly inducedrepresentation: c-ind GK σ P max ⊗ Z ˆ Z χ ∼ = ι GB ˆ δ c-ind GK σ P ⊗ Z ˆ Z χ ∼ = ι GB ˆ δ w ⊕ ι GB ˆ δ w c-ind GK σ P min ⊗ Z ˆ Z χ ∼ = ι GB ˆ δ w , where w ∈ W is the longest element and w , w ∈ W \{ , w } . Here theelements w , w are chosen such that { ι GB ˆ δ , ι GB ˆ δ w , ι GB ˆ δ w , ι GB ˆ δ w } = { ι GB ˆ δ w , w ∈ W } is the set (consisting of four pairwise non-isomorphic elements) of inducedrepresentations of the form ι GB ˆ δ w . Using compatibility with parabolic in-duction we deduce that (in the coordinates introduced above) ˆ F x = C [[ t , t , t , u , u ]] / (( t − t ) u , u ) ⊕ C [[ t , t , t , u , u ]] / ( u , ( t − t ) u ) . In other words the completion ˆ F x is the direct sum of the structure sheavesof ˆ X ˇ G, ,x ∪ ˆ Z and ˆ X ˇ G, ,x ∪ ˆ Z . References [1] T. Barnet-Lamb, T. Gee, D. Geraghty, R. Taylor, Potential automorphy and change ofweight . Ann. Math. , pp. 501-609 (2014).[2] J. 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